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# NCERT Solutions Class 7 Maths Chapter 11 Exponents and Powers

Last updated date: 11th Sep 2024
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## NCERT Solutions for Maths Class 7 Chapter 11 Exponents and Powers - FREE PDF Download

Class 7 Maths NCERT Solutions for Chapter 11 Exponents and Powers, provided by Vedantu, offer a comprehensive and straightforward explanation of key concepts. This chapter covers the fundamental rules and properties of exponents, making it crucial for understanding higher-level math topics. The solutions are designed to help students grasp the basics, such as multiplying and dividing powers, using standard form, and understanding negative exponents.

Table of Content
1. NCERT Solutions for Maths Class 7 Chapter 11 Exponents and Powers - FREE PDF Download
2. Glance on Maths Chapter 11 Class 7 - Exponents and Powers
3. Access Exercise Wise NCERT Solutions for Chapter 11 Maths Class 7
4. Exercises Under NCERT Solutions for Class 7 Maths Chapter 11 Exponents and Powers
5. Access NCERT Solutions for Class 7 Maths Chapter 11 – Exponents and Powers
5.1Exercise 11.1
6. Class 7 Maths Chapter 11: Exercises Breakdown
7. Other Study Material for CBSE Class 7 Maths Chapter 11
8. Chapter-Specific NCERT Solutions for Class 7 Maths
FAQs

The focus of this class 7 maths Ch 11 is on simplifying expressions involving exponents and powers, which is essential for solving complex mathematical problems efficiently. Students should pay particular attention to the laws of exponents, as these form the foundation for many algebraic operations. Access the latest CBSE Class 7 Maths Syllabus here.

## Glance on Maths Chapter 11 Class 7 - Exponents and Powers

• The chapter focuses on understanding exponents and powers to express large numbers in simplified form.

• Key topics include the laws of exponents such as multiplication and division of powers, power of a power, and negative exponents.

• Important concepts involve expressing numbers in standard form and scientific notation, and simplifying expressions involving exponents.

• Foundational skills are essential for algebra and higher-level math, emphasizing the mastery of rules and properties of exponents.

• 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.

• am *an = am+n

1. am / an = am-n

2. (am)n = am+n

3. a0 = 1

4. -1even number  = 1

5. -1odd number  = -1

6.  am *bm = (ab)m

• There are three exercises (17 fully solved questions) in exponents and powers class 7.

## Access Exercise Wise NCERT Solutions for Chapter 11 Maths Class 7

 S.No. Current Syllabus Exercises of Class 7 Maths Chapter 11 1 NCERT Solutions of Class 7 Maths Exponents and Powers Exercise 11.1 2 NCERT Solutions of Class 7 Maths Exponents and Powers Exercise 11.2 3 NCERT Solutions of Class 7 Maths Exponents and Powers Exercise 11.3

## Exercises Under NCERT Solutions for Class 7 Maths Chapter 11 Exponents and Powers

### Exercise 11.1

• Expressing numbers in exponential form (e.g., writing 64 as ${{\text{2}}^{\text{6}}}$)

• Identifying the base and exponent in exponential terms (e.g., recognizing 5 in ${{\text{5}}^{\text{3}}}$ as the base)

• Comparing numbers written in exponential form

### Exercise 11.2

• Simplifying exponential expressions using basic rules (like am * an = am+n)

• Evaluating powers of numbers (e.g., finding the value of ${{\text{3}}^{\text{4}}}$)

• Using order of operations with exponents

### Exercise 11.3

• Introduction to negative exponents

## Access NCERT Solutions for Class 7 Maths Chapter 11 – Exponents and Powers

### Exercise 11.1

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 11.2

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

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 11.3

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. $\text{120719}$

Ans: We have to expand $\text{120719}$.

$\text{120719 = 100000 + 20000 + 700 +10 + 9}$

$\Rightarrow \text{120719 = 1 }\!\!\times\!\!\text{ 100000 + 2 }\!\!\times\!\!\text{ 10000 + 7 }\!\!\times\!\!\text{ 100 +1 }\!\!\times\!\!\text{ 10 + 9 }\!\!\times\!\!\text{ 1}$

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

Hence, the expanded form of $\text{120719}$ 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 $\text{1}\text{.027 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{9}}}$.

## Class 7 Maths Chapter 11: Exercises Breakdown

 Exercise Number of Questions Exercise 11.1 8 Questions & Solutions Exercise 11.2 5 Questions & Solutions Exercise 11.3 4 Questions & Solutions

## Conclusion

In conclusion, Exponents and Powers Class 7 equips you with a foundational understanding of expressing large numbers compactly and efficiently. Learn how to write repeated multiplication using exponents, explore properties of exponents for simplifying calculations, and gain an introduction to working with powers. With this knowledge, you can solve problems involving large numbers and lay the groundwork for more complex mathematical concepts in higher grades. In previous years exams around 1-2 questions appeared from ​​Class 7 Maths Ch 11.

## Other Study Material for CBSE Class 7 Maths Chapter 11

 S.No. Important Links for Chapter 11 Exponents and Powers 1 Class 7 Exponents and Powers Important Questions 2 Class 7 Exponents and Powers Revision Notes 3 Class 7 Exponents and Powers Important Formulas 4 Class 7 Exponents and Powers NCERT Exemplar Solution

## Chapter-Specific NCERT Solutions for Class 7 Maths

Given below are the chapter-wise NCERT Solutions for Class 7 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

## FAQs on NCERT Solutions Class 7 Maths Chapter 11 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.

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.

10. What are the basic rules of exponents covered in exponents and powers class 7 pdf?

The basic rules of exponents include multiplying powers with the same base, dividing powers with the same base, and raising a power to another power. These rules are fundamental for simplifying complex expressions. Understanding these principles is essential for solving problems efficiently and accurately in various mathematical contexts.

11. How do exponents help in representing large numbers?

In exponents and powers class 7 pdf allow large numbers to be written in a compact form using powers, which simplifies both writing and calculations. For example, instead of writing a million, we use an exponent to express it succinctly. This method is particularly useful in scientific notation, making it easier to handle and communicate very large or very small values.

12. What is the significance of negative exponents in class 7 maths Chapter 11 pdf solutions?

In class 7 maths Chapter 11 pdf solutions negative exponents represent the reciprocal of the base raised to the corresponding positive exponent. This means that an expression with a negative exponent is equivalent to one divided by that base raised to the positive exponent. Understanding negative exponents is crucial for simplifying expressions and solving equations that involve powers and roots.