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NCERT Solutions for Class 7 Maths Chapter 13 - Exponents And Powers

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NCERT Solutions for Class 7 Maths Chapter 13 Exponents and Powers (FREE PDF Download)

NCERT Solutions for Class 7 Maths Chapter 13, Exponent, and Powers provide you with the study material in this segment. This study material will help you to get a better understanding of the concepts and solving the sums will be easy when you go through these Solutions. You will learn Exponents of natural numbers and rules of exponents through observing patterns to arrive at generalization. Every NCERT Solution is provided to make the study simple and interesting on Vedantu. Subjects like Science, Maths, English will become easy to study if you have access to NCERT Solution for Class 7 Science , Maths solutions and solutions of other subjects.


Class:

NCERT Solutions for Class 7

Subject:

Class 7 Maths

Chapter Name:

Chapter 13 - Exponents And Powers

Content-Type:

Text, Videos, Images and PDF Format

Academic Year:

2023-24

Medium:

English and Hindi

Available Materials:

  • Chapter Wise

  • Exercise Wise

Other Materials

  • Important Questions

  • Revision Notes



List of Topics Covered Under NCERT Solutions for Class 7 Maths Chapter 13 Exponents and Powers

Introduction

Exponents

Laws of Exponents

Examples of Laws of Exponents

Decimal Number system

Expressing Larger Numbers in Standard Form


A Glance About The Topic

  • We can use exponents mainly to cut short the long number. For example, 10,000 can be written as 104

  • If the bases are equal, the exponents will follow some basic laws. 

  1. am *an = am+n

  2. am / an = am-n

  3. (am)n = am+n

  4. a0 = 1

  5. -1even number  = 1

  6. -1odd number  = -1

  7.  am *bm = (ab)m

Access NCERT Solutions for Class 7 Maths Chapter 13 – Exponents and Powers

Exercise 13.1

1. Find the value of:

i. ${{\text{2}}^{\text{6}}}$

Ans: We have to find the value of ${{\text{2}}^{\text{6}}}$. It is $\text{2}$ raised to the power of $\text{6}$.

$\therefore {{\text{2}}^{\text{6}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2}$

$\Rightarrow {{\text{2}}^{\text{6}}}\text{ = 64}$

Hence, the value of ${{\text{2}}^{\text{6}}}$ is $\text{64}$.

ii. ${{\text{9}}^{\text{3}}}$

Ans: We have to find the value of ${{\text{9}}^{\text{3}}}$. It is $\text{9}$ raised to the power of $\text{3}$.

$\therefore {{\text{9}}^{\text{3}}}\text{ = 9 }\!\!\times\!\!\text{ 9 }\!\!\times\!\!\text{ 9}$

$\Rightarrow {{\text{9}}^{\text{3}}}\text{ = 729}$

Hence, the value of ${{\text{9}}^{\text{3}}}$ is $\text{729}$.

iii. $\text{1}{{\text{1}}^{\text{2}}}$

Ans: We have to find the value of $\text{1}{{\text{1}}^{\text{2}}}$. It is $\text{11}$ raised to the power of $\text{2}$.

$\therefore \text{1}{{\text{1}}^{\text{2}}}\text{ = 11 }\!\!\times\!\!\text{ 11}$

$\Rightarrow \text{1}{{\text{1}}^{\text{2}}}\text{ = 121}$

Hence, the value of $\text{1}{{\text{1}}^{2}}$ is $\text{121}$.

iv. ${{\text{5}}^{\text{4}}}$

Ans: We have to find the value of ${{\text{5}}^{\text{4}}}$. It is $\text{5}$ raised to the power of $\text{4}$.

$\therefore {{\text{5}}^{\text{4}}}\text{ = 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5}$

$\Rightarrow {{\text{5}}^{\text{4}}}\text{ = 625}$

Hence, the value of ${{\text{5}}^{\text{4}}}$ is $\text{625}$.


2. Express the following in exponential form:

i. $\text{6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6}$

Ans: We have to find the exponential form of $\text{6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6}$.

It is $\text{6}$ multiplied four times. So, it is $\text{6}$ raised to the power of $\text{4}$.

$\therefore \text{6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 = }{{\text{6}}^{\text{4}}}$

Hence, $\text{6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6}$ can be written as ${{\text{6}}^{\text{4}}}$.

ii. $\text{t }\!\!\times\!\!\text{ t}$

Ans: We have to find the exponential form of $\text{t }\!\!\times\!\!\text{ t}$.

It is $\text{t}$ multiplied two times. So, it is $\text{t}$ raised to the power of $\text{2}$.

$\therefore \text{t }\!\!\times\!\!\text{ t = }{{\text{t}}^{\text{2}}}$

Hence, $\text{t }\!\!\times\!\!\text{ t}$ can be written as ${{\text{t}}^{\text{2}}}$.

iii. $\text{b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b}$

Ans: We have to find the exponential form of $\text{b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b}$.

It is $\text{b}$ multiplied four times. So, it is $\text{b}$ raised to the power of $\text{4}$.

$\therefore \text{b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b = }{{\text{b}}^{\text{4}}}$

Hence, $\text{b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b }\!\!\times\!\!\text{ b}$ can be written as ${{\text{b}}^{\text{4}}}$.

iv. $\text{5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7}$

Ans: We have to find the exponential form of $\text{5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7}$.

It is $\text{5}$ multiplied two times and $\text{7}$ multiplied three times. So, it is $\text{5}$ raised to the power of $\text{2}$ multiplied by $\text{7}$ raised to the power of $\text{3}$.

$\therefore \text{5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7 = }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}$

Hence, $\text{5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7}$ can be written as ${{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}$.

v. $\text{2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a}$

Ans: We have to find the exponential form of $\text{2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a}$.

It is $\text{2}$ multiplied two times and $\text{a}$ multiplied two times. So, it is $\text{2}$ raised to the power of $\text{2}$ multiplied by $\text{a}$ raised to the power of $\text{2}$.

$\therefore \text{2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a = }{{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}$

Hence, $\text{2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a}$ can be written as ${{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}$.

vi. $\text{a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ d}$

Ans: We have to find the exponential form of $\text{a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ d}$.

It is $\text{a}$ multiplied three times and $\text{c}$ multiplied four times and $\text{d}$ multiplied once.

$\therefore \text{a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ d = }{{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{c}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ d}$

Hence, $\text{a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ a }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ c }\!\!\times\!\!\text{ d}$ can be written as ${{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{c}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ d}$.


3. Express each of the following numbers using exponential notation:

i. $\text{512}$

Ans: We can write $\text{512}$ as following:

$\text{512 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2}$

$\Rightarrow \text{512 = }{{\text{2}}^{\text{9}}}$

Hence, the value of $\text{512}$ in exponential form is ${{\text{2}}^{\text{9}}}$.

ii. $\text{343}$

Ans: We can write $\text{343}$ as following:

$\text{343 = 7 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 7}$

$\Rightarrow \text{343 = }{{\text{7}}^{\text{3}}}$

Hence, the value of $\text{343}$ in exponential form is ${{\text{7}}^{\text{3}}}$.

iii. $\text{729}$

Ans: We can write $\text{729}$ as following:

$\text{729 = 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3}$

$\Rightarrow \text{729 = }{{\text{3}}^{\text{6}}}$

Hence, the value of $\text{729}$ in exponential form is ${{\text{3}}^{\text{6}}}$.

iv. $\text{3125}$

Ans: We can write $\text{3125}$ as following:

$\text{3125 = 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5}$

$\Rightarrow \text{3125 = }{{\text{5}}^{\text{5}}}$

Hence, the value of $\text{3125}$ in exponential form is ${{\text{5}}^{\text{5}}}$.


4. Identify the greater number, wherever possible, in each of the following:

i. ${{\text{4}}^{\text{3}}}$ and ${{\text{3}}^{\text{4}}}$

Ans: We will first write the values of each of the exponential forms.

${{\text{4}}^{\text{3}}}\text{ = 4 }\!\!\times\!\!\text{ 4 }\!\!\times\!\!\text{ 4 = 64}$

${{\text{3}}^{\text{4}}}\text{ = 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 = 81}$

Since $\text{81 > 64}$, we can say that ${{\text{3}}^{\text{4}}}\text{ > }{{\text{4}}^{\text{3}}}$.

Hence, the greater number is ${{\text{3}}^{\text{4}}}$.

ii. ${{\text{5}}^{\text{3}}}$ or ${{\text{3}}^{\text{5}}}$

Ans: We will first write the values of each of the exponential forms.

${{\text{5}}^{\text{3}}}\text{ = 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5 = 125}$

${{\text{3}}^{\text{5}}}\text{ = 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 = 243}$

Since $\text{125 < 243}$, we can say that ${{\text{3}}^{\text{5}}}\text{ > }{{\text{5}}^{\text{3}}}$.

Hence, the greater number is ${{\text{3}}^{\text{5}}}$.

iii. ${{\text{2}}^{\text{8}}}$ or ${{\text{8}}^{\text{2}}}$

Ans: We will first write the values of each of the exponential forms.

${{\text{2}}^{\text{8}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 = 256}$

${{\text{8}}^{\text{2}}}\text{ = 8 }\!\!\times\!\!\text{ 8 = 64}$

Since $\text{256  64}$, we can say that ${{\text{2}}^{\text{8}}}\text{  }{{\text{8}}^{\text{2}}}$.

Hence, the greater number is ${{\text{2}}^{\text{8}}}$.

iv. $\text{10}{{\text{0}}^{\text{2}}}$ or ${{\text{2}}^{\text{100}}}$

Ans: We will first write the values of each of the exponential forms.

$\text{10}{{\text{0}}^{\text{2}}}\text{ = 100 }\!\!\times\!\!\text{ 100 = 10000}$

${{\text{2}}^{\text{100}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ }...\text{14 times  }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ }...$

$\Rightarrow {{\text{2}}^{\text{100}}}\text{ = 16384 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2}\times ...$

Since $\text{10000  < 16384 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ }...$, we can say that ${{\text{2}}^{\text{100}}}\text{  >10}{{\text{0}}^{\text{2}}}$.

Hence, the greater number is ${{\text{2}}^{\text{100}}}$.

v. ${{\text{2}}^{\text{10}}}$ or $\text{1}{{\text{0}}^{\text{2}}}$

Ans: We will first write the values of each of the exponential forms.

${{\text{2}}^{\text{10}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 = 1024}$

$\text{1}{{\text{0}}^{\text{2}}}\text{ = 10 }\!\!\times\!\!\text{ 10 = 100}$

Since $\text{1024  100}$, we can say that ${{\text{2}}^{\text{10}}}\text{  1}{{\text{0}}^{\text{2}}}$.

Hence, the greater number is ${{\text{2}}^{\text{10}}}$.


5. Express each of the following as product of their prime factors:

i. $\text{648}$

Ans: We can write $\text{648}$ as following:

$\text{648 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3}$

$\Rightarrow \text{648 = }{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}$

Hence, the value of $\text{648}$ in exponential form is ${{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}$.

ii. $\text{405}$

Ans: We can write $\text{405}$ as following:

$\text{405 = 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 5}$

$\Rightarrow \text{405 = }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ 5}$

Hence, the value of $\text{405}$ in exponential form is $\text{5 }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}$.

iii. $\text{540}$

Ans: We can write $\text{540}$ as following:

$\text{540 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 5}$

$\Rightarrow \text{540 = }{{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5}$

Hence, the value of $\text{540}$ in exponential form is ${{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5}$.

iv. $\text{3600}$

Ans: We can write $\text{3600}$ as following:

$\text{3600 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 5}$

$\Rightarrow \text{3600 = }{{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}$

Hence, the value of $\text{3600}$ in exponential form is ${{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}$.


6. Simplify:

i. $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}$

Ans: We have to simplify $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}$.

$\therefore \text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ = 2 }\!\!\times\!\!\text{ 10 }\!\!\times\!\!\text{ 10 }\!\!\times\!\!\text{ 10}$

$\Rightarrow \text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ = 2000}$

Hence, the value of $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}$ is $\text{2000}$.

ii. ${{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}$

Ans: We have to simplify ${{\text{7}}^{2}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}$.

\[\therefore {{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}\text{ = 7 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2}\]

$\Rightarrow {{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}\text{ = 196}$

Hence, the value of ${{\text{7}}^{2}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}$ is $\text{196}$.

iii. ${{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5}$

Ans: We have to simplify ${{\text{2}}^{3}}\text{ }\!\!\times\!\!\text{ 5}$.

\[\therefore {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 5}\]

$\Rightarrow {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5 = 40}$

Hence, the value of ${{\text{2}}^{3}}\text{ }\!\!\times\!\!\text{ 5}$ is $\text{40}$.

iv. $\text{3 }\!\!\times\!\!\text{ }{{\text{4}}^{\text{4}}}$

Ans: We have to simplify $\text{3 }\!\!\times\!\!\text{ }{{\text{4}}^{\text{4}}}$.

\[\therefore \text{3 }\!\!\times\!\!\text{ }{{\text{4}}^{\text{4}}}\text{ = 3 }\!\!\times\!\!\text{ 4 }\!\!\times\!\!\text{ 4 }\!\!\times\!\!\text{ 4 }\!\!\times\!\!\text{ 4}\]

$\Rightarrow \text{3 }\!\!\times\!\!\text{ }{{\text{4}}^{\text{4}}}\text{ = 768}$

Hence, the value of $\text{3 }\!\!\times\!\!\text{ }{{\text{4}}^{\text{4}}}$ is $\text{768}$.

v. $\text{0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}$

Ans: We have to simplify $\text{0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}$.

\[\therefore \text{0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{2}}\text{ = 0 }\!\!\times\!\!\text{ 10 }\!\!\times\!\!\text{ 10}\]

$\Rightarrow \text{0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ = 0}$

Hence, the value of $\text{0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}$ is $\text{0}$.

vi. ${{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}$

Ans: We have to simplify ${{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}$.

\[\therefore {{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ = 5 }\!\!\times\!\!\text{ 5 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3}\]

$\Rightarrow {{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ = 675}$

Hence, the value of ${{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}$ is $\text{675}$.

vii. ${{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}$

Ans: We have to simplify ${{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}$.

\[\therefore {{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3}\]

$\Rightarrow {{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}\text{ = 144}$

Hence, the value of ${{\text{2}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}$ is $\text{144}$.

viii. ${{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}$

Ans: We have to simplify ${{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}$.

\[\therefore {{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ = 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 10 }\!\!\times\!\!\text{ 10 }\!\!\times\!\!\text{ 10 }\!\!\times\!\!\text{ 10}\]

$\Rightarrow {{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ = 90000}$

Hence, the value of ${{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}$ is $\text{90000}$.


7. Simplify:

i. ${{\left( \text{-4} \right)}^{\text{3}}}$

Ans: We have to simplify ${{\left( \text{-4} \right)}^{\text{3}}}$.

\[\therefore {{\left( \text{-4} \right)}^{\text{3}}}\text{ = }\left( \text{-4} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-4} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-4} \right)\]

$\Rightarrow {{\left( \text{-4} \right)}^{\text{3}}}\text{ = -64}$

Hence, the value of ${{\left( \text{-4} \right)}^{\text{3}}}$ is $\text{-64}$.

ii. $\left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }{{\left( \text{-2} \right)}^{\text{3}}}$

Ans: We have to simplify $\left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }{{\left( \text{-2} \right)}^{\text{3}}}$.

\[\therefore \left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }{{\left( \text{-2} \right)}^{\text{3}}}\text{ = }\left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-2} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-2} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-2} \right)\]

$\Rightarrow \left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }{{\left( \text{-2} \right)}^{\text{3}}}\text{ = 24}$

Hence, the value of $\left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }{{\left( \text{-2} \right)}^{\text{3}}}$ is $\text{24}$.

iii. ${{\left( \text{-3} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-5} \right)}^{\text{2}}}$

Ans: We have to simplify ${{\left( \text{-3} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-5} \right)}^{\text{2}}}$.

\[\therefore {{\left( \text{-3} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-5} \right)}^{\text{2}}}\text{ = }\left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-3} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-5} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-5} \right)\]

$\Rightarrow {{\left( \text{-3} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-5} \right)}^{\text{2}}}\text{ = 225}$

Hence, the value of ${{\left( \text{-3} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-5} \right)}^{\text{2}}}$ is $\text{225}$.

iv. ${{\left( \text{-2} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-10} \right)}^{\text{3}}}$

Ans: We have to simplify ${{\left( \text{-2} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-10} \right)}^{\text{3}}}$.

\[\therefore {{\left( \text{-2} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-10} \right)}^{\text{3}}}\text{ = }\left( \text{-2} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-2} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-2} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-10} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-10} \right)\text{ }\!\!\times\!\!\text{ }\left( \text{-10} \right)\]

$\Rightarrow {{\left( \text{-2} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-10} \right)}^{\text{3}}}\text{ = 8000}$

Hence, the value of ${{\left( \text{-2} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{-10} \right)}^{\text{3}}}$ is $\text{8000}$.


8. Compare the following numbers:

i. $\text{2}\text{.7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{12}}}\text{; 1}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}$

Ans: We have to compare $\text{2}\text{.7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{12}}}$ and $\text{1}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}$.

We will get the solution by simply comparing the exponents of base $\text{10}$.

$\therefore \text{1}{{\text{0}}^{\text{12}}}\text{  1}{{\text{0}}^{\text{8}}}$

Hence, we can say that $\text{2}\text{.7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{12}}}\text{  1}\text{.5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}$.

ii. $\text{4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{14}}}\text{; 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{17}}}$

Ans: We have to compare $\text{4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{14}}}$ and $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{17}}}$.

We will get the solution by simply comparing the exponents of base $\text{10}$.

$\therefore \text{1}{{\text{0}}^{\text{17}}}\text{  1}{{\text{0}}^{\text{14}}}$

Hence, we can say that $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{17}}}\text{  4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{14}}}$.


Exercise 13.2

1. Using laws of exponents, simplify and write the answer in exponential form:

i. ${{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{8}}}$

Ans: We have to simplify ${{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{8}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\therefore {{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{8}}}\text{ = }{{\text{3}}^{\text{2+4+8}}}$

$\Rightarrow {{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{8}}}\text{ = }{{\text{3}}^{\text{14}}}$

Hence, we can write ${{\text{3}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{8}}}$ as ${{\text{3}}^{\text{14}}}$.

ii. ${{\text{6}}^{\text{15}}}\text{ }\!\!\div\!\!\text{ }{{\text{6}}^{\text{10}}}$

Ans: We have to simplify ${{\text{6}}^{\text{15}}}\text{ }\!\!\div\!\!\text{ }{{\text{6}}^{\text{10}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\therefore {{\text{6}}^{\text{15}}}\text{ }\!\!\div\!\!\text{ }{{\text{6}}^{\text{10}}}\text{ = }{{\text{6}}^{\text{15-10}}}$

$\Rightarrow {{\text{6}}^{\text{15}}}\text{ }\!\!\div\!\!\text{ }{{\text{6}}^{\text{10}}}\text{ = }{{\text{6}}^{\text{5}}}$

Hence, we can write ${{\text{6}}^{\text{15}}}\text{ }\!\!\div\!\!\text{ }{{\text{6}}^{\text{10}}}$ as ${{\text{6}}^{\text{5}}}$.

iii. ${{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}$

Ans: We have to simplify ${{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\therefore {{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}\text{ = }{{\text{a}}^{\text{3+2}}}$

$\Rightarrow {{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}\text{ = }{{\text{a}}^{\text{5}}}$

Hence, we can write ${{\text{a}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}$ as ${{\text{a}}^{\text{5}}}$.

iv. ${{\text{7}}^{\text{x}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}$

Ans: We have to simplify ${{\text{7}}^{\text{x}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\therefore {{\text{7}}^{\text{x}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ = }{{\text{7}}^{\text{x+2}}}$

$\Rightarrow {{\text{7}}^{\text{x}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ = }{{\text{7}}^{\text{x+2}}}$

Hence, we can write ${{\text{7}}^{\text{x}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}$ as ${{\text{7}}^{\text{x+2}}}$.

v. ${{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$

Ans: We have to simplify ${{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$.

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\therefore {{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{2 }\!\!\times\!\!\text{ 3}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{6}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\therefore {{\text{5}}^{\text{6}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{6-3}}}$

$\Rightarrow {{\text{5}}^{\text{6}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{3}}}$

Hence, we can write ${{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$ as ${{\text{5}}^{\text{3}}}$.

vi. ${{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}$

Ans: We have to simplify ${{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{m}}}\text{ = }{{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{m}}}$.

$\therefore {{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}\text{ = }{{\left( \text{2 }\!\!\times\!\!\text{ 5} \right)}^{\text{5}}}$

$\Rightarrow {{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}\text{ = 1}{{\text{0}}^{\text{5}}}$

Hence, we can write ${{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}$ as $\text{1}{{\text{0}}^{\text{5}}}$.

vii. ${{\text{a}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{4}}}$

Ans: We have to simplify ${{\text{a}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{4}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{m}}}\text{ = }{{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{m}}}$.

$\therefore {{\text{a}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{4}}}\text{ = }{{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{4}}}$

Hence, we can write ${{\text{a}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{4}}}$ as ${{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{4}}}$.

viii. ${{\left( {{\text{3}}^{\text{4}}} \right)}^{\text{3}}}$

Ans: We have to simplify ${{\left( {{\text{3}}^{\text{4}}} \right)}^{\text{3}}}$.

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\therefore {{\left( {{\text{3}}^{\text{4}}} \right)}^{\text{3}}}\text{ = }{{\text{3}}^{\text{4 }\!\!\times\!\!\text{ 3}}}\text{ = }{{\text{3}}^{\text{12}}}$

Hence, we can write ${{\left( {{\text{3}}^{\text{4}}} \right)}^{\text{3}}}$ as ${{\text{3}}^{\text{12}}}$.

ix. $\left( {{\text{2}}^{\text{20}}}\text{ }\!\!\div\!\!\text{ }{{\text{2}}^{\text{15}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}$

Ans: We have to simplify $\left( {{\text{2}}^{\text{20}}}\text{ }\!\!\div\!\!\text{ }{{\text{2}}^{\text{15}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\therefore \left( {{\text{2}}^{\text{20}}}\text{ }\!\!\div\!\!\text{ }{{\text{2}}^{\text{15}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ = }\left( {{\text{2}}^{\text{20-15}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ = }{{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{m}}}\text{ = }{{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{m}}}$.

$\therefore {{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ = }{{\text{2}}^{\text{5+3}}}\text{ = }{{\text{2}}^{\text{8}}}$

Hence, we can write $\left( {{\text{2}}^{\text{20}}}\text{ }\!\!\div\!\!\text{ }{{\text{2}}^{\text{15}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}$ as ${{\text{2}}^{\text{8}}}$.

x. ${{\text{8}}^{\text{t}}}\text{ }\!\!\div\!\!\text{ }{{\text{8}}^{\text{2}}}$

Ans: We have to simplify ${{\text{8}}^{\text{t}}}\text{ }\!\!\div\!\!\text{ }{{\text{8}}^{\text{2}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\therefore {{\text{8}}^{\text{t}}}\text{ }\!\!\div\!\!\text{ }{{\text{8}}^{\text{2}}}\text{ = }{{\text{8}}^{\text{t-2}}}$

Hence, we can write ${{\text{8}}^{\text{t}}}\text{ }\!\!\div\!\!\text{ }{{\text{8}}^{\text{2}}}$ as ${{\text{8}}^{\text{t-2}}}$.


2. Simplify and express each of the following in exponential form:

i. $\dfrac{{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ 4}}{\text{3 }\!\!\times\!\!\text{ 32}}$

Ans: We have to simplify $\dfrac{{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ 4}}{\text{3 }\!\!\times\!\!\text{ 32}}$.

$\therefore \dfrac{{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ 4}}{\text{3 }\!\!\times\!\!\text{ 32}}\text{ = }\dfrac{{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}}{\text{3 }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}$  

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\Rightarrow \dfrac{{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{2}}}}{\text{3 }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }\dfrac{{{\text{2}}^{\text{3+2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}}{\text{3 }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }\dfrac{{{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}}{\text{3 }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }\dfrac{{{\text{3}}^{\text{4}}}}{\text{3}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \dfrac{{{\text{3}}^{\text{4}}}}{\text{3}}\text{ = }{{\text{3}}^{\text{4-1}}}\text{ = }{{\text{3}}^{\text{3}}}$

Hence, we can write $\dfrac{{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ 4}}{\text{3 }\!\!\times\!\!\text{ 32}}$ as ${{\text{3}}^{\text{3}}}$.

ii. $\left[ {{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}$

Ans: We have to simplify $\left[ {{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}$.

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\therefore \left[ {{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ = }\left[ {{\text{5}}^{\text{2 }\!\!\times\!\!\text{ 3}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ = }\left[ {{\text{5}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\Rightarrow \left[ {{\text{5}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ = }{{\text{5}}^{\text{6+4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ = }{{\text{5}}^{\text{10}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow {{\text{5}}^{\text{10}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ = }{{\text{5}}^{\text{10-7}}}\text{ = }{{\text{5}}^{\text{3}}}$

Hence, we can write $\left[ {{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{4}}} \right]\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{7}}}$ as ${{\text{5}}^{\text{3}}}$.

iii. $\text{2}{{\text{5}}^{\text{4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$

Ans: We have to simplify $\text{2}{{\text{5}}^{\text{4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$.

$\therefore \text{2}{{\text{5}}^{\text{4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\Rightarrow {{\left( {{\text{5}}^{\text{2}}} \right)}^{\text{4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{2 }\!\!\times\!\!\text{ 4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{8}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow {{\text{5}}^{\text{8}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}\text{ = }{{\text{5}}^{\text{8-3}}}\text{ = }{{\text{5}}^{\text{5}}}$

Hence, we can write $\text{2}{{\text{5}}^{\text{4}}}\text{ }\!\!\div\!\!\text{ }{{\text{5}}^{\text{3}}}$ as ${{\text{5}}^{\text{5}}}$.

iv. $\dfrac{\text{3 }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{8}}}}{\text{21 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{3}}}}$

Ans: We have to simplify $\dfrac{\text{3 }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{8}}}}{\text{21 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{3}}}}$.

$\therefore \dfrac{\text{3 }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{8}}}}{\text{21 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{3}}}}\text{ = }\dfrac{\text{3 }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{8}}}}{\text{3 }\!\!\times\!\!\text{ 7 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{3}}}}\text{ = }\dfrac{\text{3}}{\text{3}}\text{ }\!\!\times\!\!\text{ }\dfrac{{{\text{7}}^{\text{2}}}}{\text{7}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{1}{{\text{1}}^{\text{8}}}}{\text{1}{{\text{1}}^{\text{3}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \dfrac{\text{3}}{\text{3}}\text{ }\!\!\times\!\!\text{ }\dfrac{{{\text{7}}^{\text{2}}}}{\text{7}}\text{ }\!\!\times\!\!\text{ }\dfrac{\text{1}{{\text{1}}^{\text{8}}}}{\text{1}{{\text{1}}^{\text{3}}}}\text{ = }{{\text{3}}^{\text{1-1}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2-1}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{8-3}}}\text{ = 7 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{5}}}$

Hence, we can write $\dfrac{\text{3 }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{8}}}}{\text{21 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{3}}}}$ as $\text{7 }\!\!\times\!\!\text{ 1}{{\text{1}}^{\text{5}}}$.

v. $\dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}}$

Ans: We have to simplify $\dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\Rightarrow \dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}}\text{ = }\dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{4+3}}}}\text{ = }\dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{7}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{7}}}}\text{ = }{{\text{3}}^{\text{7-7}}}\text{ = }{{\text{3}}^{\text{0}}}\text{ = 1}$

Hence, we can write $\dfrac{{{\text{3}}^{\text{7}}}}{{{\text{3}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}}$ as $\text{1}$.

vi. ${{\text{2}}^{\text{0}}}\text{+}{{\text{3}}^{\text{0}}}\text{+}{{\text{4}}^{ \text{0}}}$

Ans: We have to simplify ${{\text{2}}^{\text{0}}}\text{+}{{\text{3}}^{\text{0}}}\text{+}{{\text{4}}^{\text{0}}}$.

We know the value of ${{\text{a}}^{\text{0}}}\text{=1}$.

$\therefore {{\text{2}}^{\text{0}}}\text{+}{{\text{3}}^{\text{0}}}\text{+}{{\text{4}}^{\text{0}}}\text{ = 1+1+1 = 3}$

Hence, we can write ${{\text{2}}^{\text{0}}}\text{+}{{\text{3}}^{\text{0}}}\text{+}{{\text{4}}^{\text{0}}}$ as $\text{3}$.

vii. ${{\text{2}}^{\text{0}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{0}}}\text{ }\!\!\times\!\!\text{ }{{\text{4}}^{\text{0}}}$

Ans: We have to simplify ${{\text{2}}^{\text{0}}}\times {{\text{3}}^{\text{0}}}\times {{\text{4}}^{\text{0}}}$.

We know the value of ${{\text{a}}^{\text{0}}}\text{=1}$.

$\therefore {{\text{2}}^{\text{0}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{0}}}\text{ }\!\!\times\!\!\text{ }{{\text{4}}^{\text{0}}}\text{ = 1 }\!\!\times\!\!\text{ 1 }\!\!\times\!\!\text{ 1 = 1}$

Hence, we can write ${{\text{2}}^{\text{0}}}\times {{\text{3}}^{\text{0}}}\times {{\text{4}}^{\text{0}}}$ as $\text{1}$.

viii. $\left( {{\text{3}}^{\text{0}}}\text{+}{{\text{2}}^{\text{0}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{0}}}$

Ans: We have to simplify $\left( {{\text{3}}^{\text{0}}}\text{+}{{\text{2}}^{\text{0}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{0}}}$.

We know the value of ${{\text{a}}^{\text{0}}}\text{=1}$.

$\therefore \left( {{\text{3}}^{\text{0}}}\text{+}{{\text{2}}^{\text{0}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{0}}}\text{ = }\left( \text{1+1} \right)\text{ }\!\!\times\!\!\text{ 1 = 2 }\!\!\times\!\!\text{ 1 = 2}$

Hence, we can write $\left( {{\text{3}}^{\text{0}}}\text{+}{{\text{2}}^{\text{0}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{0}}}$ as $\text{2}$.

ix. $\dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\text{4}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}$

Ans: We have to simplify $\dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\text{4}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}$.

$\therefore \dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\text{4}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}\text{ = }\dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\left( {{\text{2}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}$

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\Rightarrow \dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\left( {{\text{2}}^{\text{2}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}\text{ = }\dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\text{2}}^{\text{2 }\!\!\times\!\!\text{ 3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}\text{ = }\dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\text{2}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}\text{ = }\dfrac{{{\text{2}}^{\text{8}}}}{{{\text{2}}^{\text{6}}}}\text{ }\!\!\times\!\!\text{ }\dfrac{{{\text{a}}^{\text{5}}}}{{{\text{a}}^{\text{3}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \dfrac{{{\text{2}}^{\text{8}}}}{{{\text{2}}^{\text{6}}}}\text{ }\!\!\times\!\!\text{ }\dfrac{{{\text{a}}^{\text{5}}}}{{{\text{a}}^{\text{3}}}}\text{ = }{{\text{2}}^{\text{8-6}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5-3}}}\text{ = }{{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{m}}}\text{ = }{{\left( \text{ab} \right)}^{\text{m}}}$.

$\Rightarrow {{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{2}}}\text{ = }{{\left( \text{2a} \right)}^{\text{2}}}$

Hence, we can write $\dfrac{{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}}{{{\text{4}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{3}}}}$ as ${{\left( \text{2a} \right)}^{\text{2}}}$.

x. $\left( \dfrac{{{\text{a}}^{\text{5}}}}{{{\text{a}}^{\text{3}}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}$

Ans: We have to simplify $\left( \dfrac{{{\text{a}}^{\text{5}}}}{{{\text{a}}^{\text{3}}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \left( \dfrac{{{\text{a}}^{\text{5}}}}{{{\text{a}}^{\text{3}}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}\text{ = }{{\text{a}}^{\text{5-3}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}\text{ = }{{\text{a}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\Rightarrow {{\text{a}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}\text{ = }{{\text{a}}^{\text{2+8}}}\text{ = }{{\text{a}}^{\text{10}}}$

Hence, we can write $\left( \dfrac{{{\text{a}}^{\text{5}}}}{{{\text{a}}^{\text{3}}}} \right)\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}$ as ${{\text{a}}^{\text{10}}}$.

xi. $\dfrac{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}{{\text{b}}^{\text{3}}}}{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}{{\text{b}}^{\text{2}}}}$

Ans: We have to simplify $\dfrac{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}{{\text{b}}^{\text{3}}}}{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}{{\text{b}}^{\text{2}}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \dfrac{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}{{\text{b}}^{\text{3}}}}{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}{{\text{b}}^{\text{2}}}}\text{ = }{{\text{4}}^{\text{5-5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8-5}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{3-2}}}\text{ = }{{\text{4}}^{\text{0}}}{{\text{a}}^{\text{3}}}{{\text{b}}^{\text{1}}}\text{ = }{{\text{a}}^{\text{3}}}\text{b}$

Hence, we can write $\dfrac{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{8}}}{{\text{b}}^{\text{3}}}}{{{\text{4}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{5}}}{{\text{b}}^{\text{2}}}}$ as ${{\text{a}}^{\text{3}}}\text{b}$.

xii. ${{\left( {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 2} \right)}^{\text{2}}}$

Ans: We have to simplify ${{\left( {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 2} \right)}^{\text{2}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\Rightarrow {{\left( {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 2} \right)}^{\text{2}}}\text{ = }{{\left( {{\text{2}}^{\text{3+1}}} \right)}^{\text{2}}}\text{ = }{{\left( {{\text{2}}^{\text{4}}} \right)}^{\text{2}}}$

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\Rightarrow {{\left( {{\text{2}}^{\text{4}}} \right)}^{\text{2}}}\text{ = }{{\text{2}}^{\text{4 }\!\!\times\!\!\text{ 2}}}\text{ = }{{\text{2}}^{\text{8}}}$

Hence, we can write ${{\left( {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 2} \right)}^{\text{2}}}$ as ${{\text{2}}^{\text{8}}}$.


3. Say true or false and justify your answer:

i. $\text{10 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{11}}}\text{ = 10}{{\text{0}}^{\text{11}}}$

Ans: The given statement is false.

Explanation:

LHS: It is given $\text{10 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{11}}}$.

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\therefore \text{10 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{11}}}\text{ = 1}{{\text{0}}^{\text{1+11}}}\text{ = 1}{{\text{0}}^{\text{12}}}$


RHS: It is given $\text{10}{{\text{0}}^{\text{11}}}$.

We have obtained that $\text{LHS }\ne \text{ RHS}$

Hence, the statement is false.

ii. ${{\text{2}}^{\text{3}}}\text{  }{{\text{5}}^{\text{2}}}$

Ans: The given statement is false.

Explanation:

LHS: It is given ${{\text{2}}^{\text{3}}}$.

$\therefore {{\text{2}}^{\text{3}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 = 8}$


RHS: It is given ${{\text{5}}^{\text{2}}}$.

$\therefore {{\text{5}}^{\text{2}}}\text{ = 5 }\!\!\times\!\!\text{ 5 = 25}$

We have obtained that $\text{LHS  < RHS}$

Hence, the statement is false.

iii. ${{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}\text{ = }{{\text{6}}^{\text{5}}}$

Ans: The given statement is false.

Explanation:

LHS: It is given ${{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}$.

$\therefore {{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{2}}}\text{ = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 = 72}$


RHS: It is given ${{\text{6}}^{\text{5}}}$.

$\therefore {{\text{6}}^{\text{5}}}\text{ = 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 }\!\!\times\!\!\text{ 6 = 7776}$

We have obtained that $\text{LHS }\ne \text{ RHS}$

Hence, the statement is false.

iv. ${{\text{3}}^{\text{0}}}\text{ = }{{\left( \text{1000} \right)}^{\text{0}}}$

Ans: The given statement is true.

Explanation:

LHS: It is given ${{\text{3}}^{\text{0}}}$.

$\therefore {{\text{3}}^{\text{0}}}\text{ = 1}$


RHS: It is given $\text{100}{{\text{0}}^{\text{0}}}$.

$\therefore \text{100}{{\text{0}}^{\text{0}}}\text{ = 1}$

We have obtained that $\text{LHS = RHS}$

Hence, the statement is true.


4. Express each of the following as a product of prime factors only in exponential form:

i. $\text{108 }\!\!\times\!\!\text{ 192}$

Ans: We have to express $\text{108 }\!\!\times\!\!\text{ 192}$ in exponential form.

$\text{108 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 = }{{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}$

$\text{192 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 = }{{\text{2}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{1}}}$

$\therefore \text{108 }\!\!\times\!\!\text{ 192 = }\left( {{\text{2}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}} \right)\text{ }\!\!\times\!\!\text{ }\left( {{\text{2}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{1}}} \right)$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\therefore \text{108 }\!\!\times\!\!\text{ 192 = }{{\text{2}}^{\text{2+6}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3+1}}}$

$\Rightarrow \text{108 }\!\!\times\!\!\text{ 192 = }{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}$

Hence, we can write $\text{108 }\!\!\times\!\!\text{ 192}$ as ${{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{4}}}$.

ii. $\text{270}$

Ans: We have to express $\text{270}$ in exponential form.

$\text{270 = 2 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 5 = }{{\text{2}}^{\text{1}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5 = 2 }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5}$

Hence, we can write $\text{270}$ as $\text{2 }\!\!\times\!\!\text{ }{{\text{3}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 5}$.

iii. $\text{729 }\!\!\times\!\!\text{ 64}$

Ans: We have to express $\text{729 }\!\!\times\!\!\text{ 64}$ in exponential form.

$\text{729 = 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 3 = }{{\text{3}}^{\text{6}}}$

$\text{64 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 = }{{\text{2}}^{\text{6}}}$

$\therefore \text{729 }\!\!\times\!\!\text{ 64 = }{{\text{3}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{6}}}$


Hence, we can write $\text{729 }\!\!\times\!\!\text{ 64}$ as ${{\text{3}}^{\text{6}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{6}}}$.

iv. $\text{768}$

Ans: We have to express $\text{768}$ in exponential form.

$\text{270 = 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 3 = }{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{1}}}\text{ = }{{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ 3}$

Hence, we can write $\text{768}$ as ${{\text{2}}^{\text{8}}}\text{ }\!\!\times\!\!\text{ 3}$.


5. Simplify:

i. $\dfrac{{{\left( {{\text{2}}^{\text{5}}} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{8}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 7}}$

Ans: We have to simplify $\dfrac{{{\left( {{\text{2}}^{\text{5}}} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{8}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 7}}$

$\dfrac{{{\left( {{\text{2}}^{\text{5}}} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{8}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 7}}\text{ = }\dfrac{{{\left( {{\text{2}}^{\text{5}}} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\left( {{\text{2}}^{\text{3}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 7}}$

We know the law of exponents ${{\left( {{\text{a}}^{\text{m}}} \right)}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m }\!\!\times\!\!\text{ n}}}$.

$\dfrac{{{\left( {{\text{2}}^{\text{5}}} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\left( {{\text{2}}^{\text{3}}} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 7}}\text{ = }\dfrac{{{\text{2}}^{\text{5 }\!\!\times\!\!\text{ 2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{2}}^{\text{3 }\!\!\times\!\!\text{ 3}}}\text{ }\!\!\times\!\!\text{ 7}}\text{ = }\dfrac{{{\text{2}}^{\text{10}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{2}}^{\text{9}}}\text{ }\!\!\times\!\!\text{ 7}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\dfrac{{{\text{2}}^{\text{10}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{2}}^{\text{9}}}\text{ }\!\!\times\!\!\text{ 7}}\text{ = }{{\text{2}}^{\text{10-9}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3-1}}}\text{ = }{{\text{2}}^{\text{1}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{2}}}\text{ =98}$

Therefore, the value of $\dfrac{{{\left( {{\text{2}}^{\text{5}}} \right)}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{7}}^{\text{3}}}}{{{\text{8}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 7}}$ is $\text{98}$.

ii. $\dfrac{\text{25 }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{\text{1}{{\text{0}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}$

Ans: We have to simplify $\dfrac{\text{25 }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{\text{1}{{\text{0}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}$.

$\dfrac{\text{25 }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{\text{1}{{\text{0}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}=\dfrac{{{\text{5}}^{2}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\left( 5\times 2 \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{m}}}\text{ = }{{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{m}}}$.

\[\Rightarrow \dfrac{{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\left( \text{5 }\!\!\times\!\!\text{ 2} \right)}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}\text{=}\dfrac{{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\text{5}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}\]

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

\[\Rightarrow \dfrac{{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\text{5}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}\text{ = }\dfrac{{{\text{5}}^{\text{2+2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\text{5}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}\text{=}\dfrac{{{\text{5}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\text{5}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}\]

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

\[\Rightarrow \dfrac{{{\text{5}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{{{\text{5}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}\text{ = }\dfrac{{{\text{5}}^{\text{4-3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8-4}}}}{{{\text{2}}^{\text{3}}}}\text{ = }\dfrac{{{\text{5}}^{\text{1}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}{{{\text{2}}^{\text{3}}}}\text{ = }\dfrac{\text{5}{{\text{t}}^{\text{4}}}}{\text{8}}\]

Therefore, the value of $\dfrac{\text{25 }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{8}}}}{\text{1}{{\text{0}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ }{{\text{t}}^{\text{4}}}}$ is $\dfrac{\text{5}{{\text{t}}^{\text{4}}}}{\text{8}}$.

iii. $\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 25}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{6}}^{\text{5}}}}$

Ans: We have to simplify $\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 25}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{6}}^{\text{5}}}}$.

$\therefore \dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 25}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{6}}^{\text{5}}}}\text{ = }\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{5 }\!\!\times\!\!\text{ 2} \right)}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{3 }\!\!\times\!\!\text{ 2} \right)}^{\text{5}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{b}}^{\text{m}}}\text{ = }{{\left( \text{a }\!\!\times\!\!\text{ b} \right)}^{\text{m}}}$.

$\Rightarrow \dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{5 }\!\!\times\!\!\text{ 2} \right)}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{3 }\!\!\times\!\!\text{ 2} \right)}^{\text{5}}}}\text{ = }\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\times\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m+n}}}$.

$\Rightarrow \dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{2}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{5+2}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}$

We know the law of exponents ${{\text{a}}^{\text{m}}}\text{ }\!\!\div\!\!\text{ }{{\text{a}}^{\text{n}}}\text{ = }{{\text{a}}^{\text{m-n}}}$.

$\Rightarrow \dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }{{\text{3}}^{\text{5-5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{7-7}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5-5}}}$

$\Rightarrow \dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = }{{\text{3}}^{\text{0}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{0}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{0}}}$

$\Rightarrow \dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{5}}}}\text{ = 1 }\!\!\times\!\!\text{ 1 }\!\!\times\!\!\text{ 1 = 1}$

Therefore, the value of $\dfrac{{{\text{3}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ }\!\!\times\!\!\text{ 25}}{{{\text{5}}^{\text{7}}}\text{ }\!\!\times\!\!\text{ }{{\text{6}}^{\text{5}}}}$ is $\text{1}$.

.

Exercise 13.3

1. Write the following numbers in the expanded form:

a. $\text{279404}$

Ans: We have to expand $\text{279404}$.

$\text{279404 = 200000+70000+9000+400+4}$

$\Rightarrow \text{279404 = 2 }\!\!\times\!\!\text{ 100000+7 }\!\!\times\!\!\text{ 10000 + 9 }\!\!\times\!\!\text{ 1000 + 4 }\!\!\times\!\!\text{ 100 + 4 }\!\!\times\!\!\text{ 1}$

$\Rightarrow \text{279404 = 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{+7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Hence, the expanded form of $\text{279404}$ is $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{+7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$.

b. $\text{3006194}$

Ans: We have to expand $\text{3006194}$.

$\text{3006194 = 3000000 + 6000 + 100 + 90 + 4}$

$\Rightarrow \text{3006194 = 3 }\!\!\times\!\!\text{ 1000000 + 6 }\!\!\times\!\!\text{ 1000 + 1 }\!\!\times\!\!\text{ 100 + 9 }\!\!\times\!\!\text{ 10 + 4 }\!\!\times\!\!\text{ 1}$

$\Rightarrow \text{3006194 = 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{6}}}\text{ + 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{+ 9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Hence, the expanded form of $\text{3006194}$ is $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{6}}}\text{ + 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{+ 9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$.

c. $\text{2806196}$

Ans: We have to expand $\text{2806196}$.

$\text{2806196 = 2000000 + 800000 + 6000 + 100 + 90 + 6}$

$\Rightarrow \text{2806196 = 2 }\!\!\times\!\!\text{ 1000000 + 8 }\!\!\times\!\!\text{ 100000 + 6 }\!\!\times\!\!\text{ 1000 + 1 }\!\!\times\!\!\text{ 100 + 9 }\!\!\times\!\!\text{ 10 + 6 }\!\!\times\!\!\text{ 1}$

$\Rightarrow \text{2806196 = 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{6}}}\text{ + 8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{+ 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{+ 9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Hence, the expanded form of $\text{2806196}$ is $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{6}}}\text{ + 8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{+ 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{+ 9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$.

d. $\text{120717}$

Ans: We have to expand $\text{120717}$.

$\text{120717 = 100000 + 20000 + 700 +10 + 7}$

$\Rightarrow \text{120717 = 1 }\!\!\times\!\!\text{ 100000 + 2 }\!\!\times\!\!\text{ 10000 + 7 }\!\!\times\!\!\text{ 100 +1 }\!\!\times\!\!\text{ 10 + 7 }\!\!\times\!\!\text{ 1}$

$\Rightarrow \text{120717 = 1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ + 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ +1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Hence, the expanded form of $\text{120717}$ is $\text{1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ + 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ +1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$.

e. $\text{20068}$

Ans: We have to expand $\text{20068}$.

$\text{20068 = 20000 +60 + 8}$

$\Rightarrow \text{20068 = 2 }\!\!\times\!\!\text{ 10000 +6 }\!\!\times\!\!\text{ 10 + 8 }\!\!\times\!\!\text{ 1}$

$\Rightarrow \text{20068 = 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ +6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Hence, the expanded form of $\text{20068}$ is $\text{2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ +6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$.


2. Find the number from each of the following expanded form:

a. $\text{8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Ans: We are given $\text{8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 0 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}\text{ + 5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

We will now simplify it.

$\text{= 8 }\!\!\times\!\!\text{ 10000 + 6 }\!\!\times\!\!\text{ 1000 + 0 }\!\!\times\!\!\text{ 100 + 4 }\!\!\times\!\!\text{ 10 + 5 }\!\!\times\!\!\text{ 1}$

$\text{= 80000 + 6000 + 0 + 40 + 5}$

$\text{= 86045}$

Hence, the required number is $\text{86045}$.

b. $\text{4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ + 5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Ans: We are given $\text{4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ + 5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}\text{ + 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

We will now simplify it.

$\text{= 4 }\!\!\times\!\!\text{ 100000 + 5 }\!\!\times\!\!\text{ 1000 + 3 }\!\!\times\!\!\text{ 100 + 0 }\!\!\times\!\!\text{ 10 + 2 }\!\!\times\!\!\text{ 1}$

$\text{= 400000 + 5000 + 300 + 0 + 2}$

$\text{= 405302}$

Hence, the required number is $\text{405302}$.

c. $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ +  5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

Ans: We are given $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ + 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{0}}}$

We will now simplify it.

$\text{= 3 }\!\!\times\!\!\text{ 10000 + 7 }\!\!\times\!\!\text{ 100 + 5 }\!\!\times\!\!\text{ 1}$

$\text{= 30000 + 700 + 5}$

$\text{= 30705}$

Hence, the required number is $\text{30705}$.

d. $\text{9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ + 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ +  3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}$

Ans: We are given $\text{9 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ + 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{2}}}\text{ + 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{1}}}$

We will now simplify it.

$\text{= 9 }\!\!\times\!\!\text{ 100000 + 2 }\!\!\times\!\!\text{ 100 + 3 }\!\!\times\!\!\text{ 10}$

$\text{= 900000 + 200 + 30}$

$\text{= 900230}$

Hence, the required number is $\text{900230}$.


3. Express the following numbers in standard form:

i. $\text{5,00,00,000}$

Ans: We have to write the given number in standard form.

$\text{5,00,00,000 = 5 }\!\!\times\!\!\text{ 1,00,00,000 = 5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{7}}}$

Hence, the standard form is $\text{5 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{7}}}$.

ii. $\text{70,00,000}$

Ans: We have to write the given number in standard form.

$\text{70,00,000 = 7 }\!\!\times\!\!\text{ 10,00,000 = 7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{6}}}$

Hence, the standard form is $\text{7 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{6}}}$.

iii. $\text{3,18,65,00,000}$

Ans: We have to write the given number in standard form.

$\text{3,18,65,00,000 = 31865 }\!\!\times\!\!\text{ 1,00,000 = 3}\text{.1865 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{ }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{ = 3}\text{.1865 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$

Hence, the standard form is $\text{3}\text{.1865  }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$.

iv. $\text{3,90,878}$

Ans: We have to write the given number in standard form.

$\text{3,90,878 = 3}\text{.90878 }\!\!\times\!\!\text{ 1,00,000 = 3}\text{.90878 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}$

Hence, the standard form is $\text{3}\text{.90878 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}$.

v. $\text{39087}\text{.8}$

Ans: We have to write the given number in standard form.

$\text{39087}\text{.8 = 3}\text{.90878 }\!\!\times\!\!\text{ 10,000 = 3}\text{.90878 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}$

Hence, the standard form is $\text{3}\text{.90878  }\!\!\times\!\!\text{  1}{{\text{0}}^{\text{4}}}$.

vi. $\text{3908}\text{.78}$

Ans: We have to write the given number in standard form.

$\text{3908}\text{.78 = 3}\text{.90878 }\!\!\times\!\!\text{ 1,000 = 3}\text{.90878 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{3}}}$

Hence, the standard form is $\text{3}\text{.90878  }\!\!\times\!\!\text{  1}{{\text{0}}^{\text{3}}}$.


4. Express the number appearing in the following statements in standard form:

a. The distance between Earth and Moon is $\text{384,000,000 m}$.

Ans: We have to write $\text{384,000,000}$in standard form.

$\text{384,000,000 = 3}\text{.84 }\!\!\times\!\!\text{ 100,000,000 = 3}\text{.84 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}$

Hence, the required standard form is $\text{3}\text{.84 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}\text{ m}$.

b. Speed of light in vacuum is $\text{300,000,000 m/s}$.

Ans: We have to write $\text{300,000,000}$in standard form.

$\text{300,000,000 = 3 }\!\!\times\!\!\text{ 100,000,000 = 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}$

Hence, the required standard form is $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{8}}}\text{ m/s}$.

c. Diameter of Earth is $\text{1,27,56,000 m}$.

Ans: We have to write $\text{1,27,56,000}$in standard form.

$\text{1,27,56,000 = 1}\text{.2756 }\!\!\times\!\!\text{ 1,00,00,000 = 1}\text{.2756 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{7}}}$

Hence, the required standard form is $\text{1}\text{.2756 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{7}}}\text{ m}$.

d. Diameter of Sun is $\text{1,400,000,000 m}$.

Ans: We have to write $\text{1,400,000,000}$ in standard form.

$\text{1,400,000,000 = 1}\text{.4 }\!\!\times\!\!\text{ 1,000,000,000 = 1}\text{.4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$

Hence, the required standard form is $\text{1}\text{.4 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}\text{ m}$.

e. In a galaxy there are on average $\text{100,000,000,000}$ stars.

Ans: We have to write $\text{100,000,000,000}$ in standard form.

$\text{100,000,000,000 = 1 }\!\!\times\!\!\text{ 100,000,000,000 = 1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{11}}}$

Hence, the required standard form is $\text{1 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{11}}}$ stars.

f. The universe is estimated to be about $\text{12,000,000,000}$ years old.

Ans: We have to write $\text{12,000,000,000}$ in standard form.

$\text{12,000,000,000 = 1}\text{.2 }\!\!\times\!\!\text{ 10,000,000,000 = 1}\text{.2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{10}}}$

Hence, the required standard form is $\text{1}\text{.2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{10}}}$ years.

g. The distance of the Sun from the centre of the Milky Way  galaxy is estimated to be $\text{300,000,000,000,000,000,000 m}$.

Ans: We have to write $\text{300,000,000,000,000,000,000}$ in standard form.

$\text{300,000,000,000,000,000,000 = 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{20}}}$

Hence, the required standard form is $\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{20}}}\text{ m}$.

h. $\text{60,230,000,000,000,000,000,000}$ molecules are contained in a drop of water weighing $\text{1}\text{.8 gm}$.

Ans: We have to write $\text{60,230,000,000,000,000,000,000}$ in standard form.

$\text{60,230,000,000,000,000,000,000 = 6}\text{.023 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{22}}}$

Hence, the required standard form is $\text{6}\text{.023 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{22}}}$ molecules.

i. The Earth has $\text{1,353,000,000}$ cubic km of water.

Ans: We have to write $\text{1,353,000,000}$ in standard form.

$\text{1,353,000,000 = 1}\text{.353 }\!\!\times\!\!\text{ 1,000,000,000 = 1}\text{.353 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$

Hence, the required standard form is $\text{1}\text{.353 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$ cubic km.

j. The population of India was about $\text{1,027,000,000}$ in March, $\text{2001}$.

Ans: We have to write $\text{1,027,000,000}$ in standard form.

$\text{1,027,000,000 = 1}\text{.027 }\!\!\times\!\!\text{ 1,000,000,000 = 1}\text{.027 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$

Hence, the required standard form is $\text{1}\text{.027 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$.


NCERT Solutions for Class 7 Maths Chapter 13 – Free PDF Download

Class 7 Maths Chapter 13 Includes:

Chapter 13 Exponents and Powers All Exercises in PDF Format

Exercise 13.1

8 Question and Solutions(1 short question and 7 long questions).

Exercise 13.2

5 Questions and Solutions(1 short question and 4 long questions).

Exercise 13.3

4 Questions and Solutions(1 short question and 3 long questions).


Facts

  • (-1) odd number = -1and (-1) even number = 1

  • Let ‘a’ and ‘b’ be non-zero integers and m, n are whole numbers, then

  1. am x an = am+n

  2. am ÷ an = am-n; m > n

  3. (am)n = amn

  4. am x bm = (ab)m

  5. am ÷ bm = (a/b)m

  6. a0 = 1

  • Any number expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10 is said to be in the standard form.

  • Exponents are used for writing long numbers as short notations in order to make these numbers easy to read, understand, and compare.

Let us look into some large numbers that can be written in short notations.

Very large numbers such as:

  1. Distance between Sun and Saturn = 1,433, 500, 000, 000 m.

  2. Mass of the Earth = 5,970, 000, 000, 000, 000, 000, 000, 000 kg.

  3. Distance between Saturn and Uranus = 1, 439, 000, 000, 000 m.

  4. Mass of Uranus = 86, 800, 000, 000, 000, 000, 000, 000, 000 kg.

 

If you look closely, these numbers cannot be read easily. We have to write these numbers, in short, to understand them easily. 

 

Exponents

Let a number be x and we multiply the number x by itself.

x * x = x2

x * x * x = x3

x * x * x * x = x4

x * x * x * x * x * x = x6

The number Xn is read as ‘X‘ raised to the power of ‘n’ or simply the nth power of ‘X‘.

Here, X is called the base, and n is called the exponent.

For example:

  1. 1000 = 10 x 10 x 10 = 103

The base is 10 and the exponent is 3.

  1. 10000 = 10 x 10 x 10 x 10 x 10 = 104

The base is 10 and the exponent is 4.

  1. 32 = 2 x 2 x 2 x 2 x 2 = 25

The base is 2 and the exponent is 5.

  1. 625 = 5 x 5 x 5 x 5  = 54.

The base is 5 and the exponent is 4.

 

In all these examples, 103, 104, 25, 54 are the exponential forms of the same values.

Since you have understood how to write exponents, you can now write the large numbers given above into short notations.

1. Distance between Sun and Saturn = 1,433, 500, 000, 000 m.

This large number can be written as 14335 x 108 m (where 14335 is a natural number, 10 is the base, and 8 is the exponent).

 

2. Mass of the Earth = 5,970, 000, 000, 000, 000, 000, 000, 000 kg.

Here, we can write the large number as 597 x 1022  kg (where 597 is a natural number, 10 is the base, and 22 is the exponent).

 

3. Distance between Saturn and Uranus = 1, 439, 000, 000, 000 m.

This large number can be written as 1439 x 109 m (where 1439 is a natural number, 10 is the base, and 9 is the exponent).

 

4. Mass of Uranus = 86, 800, 000, 000, 000, 000, 000, 000, 000 kg.

Similarly, we can write the large number as 868 x 1023  kg (where 868 is a natural number, 10 is the base, and 23 is the exponent).

Example 1

Express 128 as a power of 2

Solution.  We have 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27

Thus, 128 = 27

 

Example 2

Express 144 as the product of the power of prime numbers.

Solution.  144 = 2 x 2 x 2 x 2 x 3 x 3 = 24 x 32

Now you can solve the exercises in 13.1 NCERT solutions.

 

Laws of Exponents

Let us now understand the laws of exponents one by one.

I. Multiplying Powers with the Same Base

am x an = a(m+n)

where ‘a’ is any non-zero integer.

‘m’ and ‘n’ are whole numbers.

Example: 42 x 45 = 42+5 = 47

38 x 33 = 38+3 = 311

b2 x b5 = b2+5 = b7

 

II. Dividing Powers with the Same Base

am ÷ an = am-n; m > n

where ‘a’ is any non-zero integer.

‘m’ and ‘n’ are whole numbers.

m > n

Example: 57 ÷ 53 = 57-3 = 54

210 ÷ 28 = 210-8 = 22

(-8)6 ÷ (-8)3 = (-8)6-3 = (-8)3

 

III. Taking Power of a Power

(am)n = amn

where ‘a’ is any non-zero integer.

‘m’ and ‘n’ are whole numbers.

Example: (23)5 = 23 x 5 = 215

(53)2 =  53 x 2 = 56

(-63)4 = (-6) 3 x 4 = (-6)12

 

IV. Multiplying Powers with Same Exponents

am x bm = (ab)m

where ‘a’ is any non-zero integer.

‘m’ and ‘n’ are whole numbers.

Example: 23 x 33 = (2 x 3)3 = 63

52 x 62 = (5 x 6)2 = 302

(-2)4 x (-3)4 =

(−2)x(−3)

(−2)x(−3)4 = 64

 

V. Dividing the Powers with the Same Exponents

am ÷ bm = (a/b)m

where ‘a’ is any non-zero integer.

‘m’ and ‘n’ are whole numbers.

Example: 54 ÷ 34 =(5/3)4         

(-3)2 ÷ p2 = (-3/p)2

 

VI. Numbers with Exponent Zero

a0 = 1 ( for any non-zero integer ‘a’)

        or

Any number (except 0) raised to the power ( or exponent) 0 is 1.

Example: (25)0 = 1

(½)0 = 1

(-18)0 = 1

After going through the laws of exponents, now you can solve the exercise 13.2 and refer to its NCERT Solutions.


Why should you study from NCERT Solutions for Class 7 Maths Chapter 13 - Exponents and Powers?

Key Features of NCERT Solutions, These solutions are designed to help students achieve proficiency in their studies. They are crafted by experienced educators who excel in teaching Class 7 Maths. Some of the features include:


  • Comprehensive explanations for each exercise and questions, promoting a deeper understanding of the subject.

  • Clear and structured presentation for easy comprehension.

  • Accurate answers aligned with the curriculum, boosting students' confidence in their knowledge.

  • Visual aids like diagrams and illustrations to simplify complex concepts.

  • Additional tips and insights to enhance students' performance.

  • Chapter summaries for quick revision.

  • Online accessibility and downloadable resources for flexible study and revision.


Conclusion

The NCERT Solutions for Class 7 Maths Chapter 13 - Exponents and Powers, provided by Vedantu, is a valuable tool for Class 7 students. It helps introduce Maths concepts in an accessible manner. The provided solutions and explanations simplify complex ideas, making it easier for Class 7 students to understand the material. By using Vedantu's resources, Students can develop a deeper understanding of NCERT concepts. These solutions are a helpful aid for grade 7 students, empowering them to excel in their studies and develop a genuine appreciation for “Exponents and Powers”.

FAQs on NCERT Solutions for Class 7 Maths Chapter 13 - Exponents And Powers

1. What do you understand by ‘Exponents and Powers’?

When a number is multiplied to itself for a certain number of times, then the number is said to be raised to a power. For example, in the following expression, 26, the integer 2 is raised to a power of 6. To get the resultant, you have to multiply 2 to itself 6 times. Here, the integer 2 is termed as the base and the integer 6 is termed as the exponent or power of 2. The exponent is the number of times for which the base is multiplied to itself.


The concept of ‘Exponents and Powers’ is introduced in the school level academics, and students are suggested to practice the sums of this chapter as they will come across various real-life applications of this concept.

2. Can I find the NCERT Solutions for Class 7 Maths Chapter 13 online?

Yes, you can find the NCERT Solutions for Class 7 Maths Chapter 13 online, on Vedantu. These solutions are verified recommended by experts. All the sums of this chapter are solved in a step by step manner in the PDF of these NCERT solutions. The concept of Exponents and Powers is very interesting and when you go through these solved exercises, you will be able to develop a deeper understanding of the same.

3. Are the NCERT Solutions for Class 7 Maths Chapter 13 reliable study resources?

Yes, the NCERT Solutions for Class 7 Maths Chapter 13 are very reliable study resources. The in-house team of subject matter experts at Vedantu has prepared these solutions to facilitate an easy learning process for all students. Every sum is solved as per the CBSE guidelines for Class 9, so students can rely on these NCERT solutions for their exam preparation. They can solve the exercises of Exponents and Powers, on their own and compare their answers with these step-wise solutions. In this way, it will become easier for students to identify their silly mistakes and rectify them. These NCERT solutions are very reliable study resources for self-study and revision purposes before the examination.

4. Can I download the NCERT Solutions for Class 7 Maths Chapter 13 ‘Exponents and Powers’ for free?

Yes, you can download the NCERT Solutions for Class 7 Maths Chapter 13 ‘Exponents and Powers’ for absolutely free of cost from Vedantu. These NCERT solutions are available in the PDF format on our mobile application as well. Thereby, making the best-rated study resources for this Math chapter available for all students. All you need to access these NCERT solutions is internet connectivity and a digital screen. Yes, you can download these NCERT solutions even on your smartphones or tablets. So download and refer to the NCERT Solutions for Class 7 Maths Chapter 13 ‘Exponents and Powers’ right away, for an easy learning experience.

5. Why are exponents necessary?

Exponents allow you to write large numbers in a readable and concise manner. The principles and regulations constructed around exponents are included in the NCERT Solutions for Class 7 Mathematics Chapter 13 Exponents and Powers. In Mathematics, science, and geography, large numbers are frequently used. It is tough not only to read them but also to perform Mathematics with them. Exponents and powers are used to make numbers easier to read and manipulate. As a result, these NCERT solutions for Class 7 Maths Chapter 13 teach students to write long numbers in short form notation.

6. Why should I practise Chapter 13 of NCERT Solutions for Class 7 Maths Exponents and Powers?

The NCERT Solutions for Class 7 Maths Exponents and Powers Chapter 13 has been carefully crafted by NCERT's top academics, making it a valuable learning resource. They made certain that the principles were conveyed in simple terms and that all of the important topics were addressed in depth. The CBSE board also strongly encourages pupils to study from the NCERT books, making it a significant practice resource for them.

7. How can CBSE students properly use NCERT Solutions for Chapter 13 of Class 7 Maths?

Students must take it carefully throughout the chapter, moving ahead only after they have a firm grasp of the concepts in each section. When exploring new concepts, poor comprehension of earlier concepts will lead to misunderstanding. They must also practice all of the solved instances to reinforce their newly acquired knowledge. They would be able to properly use the NCERT Solutions Class 7 Maths Chapter 13 in this manner. Refer to Vedantu for the NCERT Solution of this chapter.

8. How many questions are there in Chapter 13 Exponents and Powers of the NCERT Solutions for Class 7 Maths?

There are 17 questions in Class 7 Maths Chapter 13 Exponents and Powers, with several subparts. 12 of them are quite simple because they require exponential representation, and 5 of them are a little more complicated due to the computations involved. All of the solved problems are available in pdf format on Vedantu's website. Solve those problems first, then cross-check with the answers you've already figured out.

9. What are the key laws in Chapter 13 of NCERT Solutions for Class 7 Maths?

The law of exponents is discussed in NCERT Solutions Class 7 Maths Chapter 13 and certain key laws are explained. Here are a few examples: The addition of exponents' powers results from multiplication, while the subtraction of exponents' powers results from division. Furthermore, the strength of power causes them to multiply. These are crucial hints that pupils will require when answering questions.