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

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 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

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.

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