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 $\text{3}$.

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

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

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 $\text{2}$.

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

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

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.

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

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 ${\text{a}}^{\text{3}}\times {\text{c}}^{\text{4}}\times \text{ d}$.

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

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.

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

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

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

Since $\text{10000 < 16384 }\times \text{ 2 }\times ...$, 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:

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

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

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

ii. $\text{405}$

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

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

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

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

iii. $\text{540}$

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

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

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

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

iv. $\text{3600}$

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

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

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

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

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

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

$$\therefore {\text{7}}^{\text{2}}\times {\text{2}}^{\text{2}}\text{ = 7 }\times \text{ 7 }\times \text{ 2 }\times \text{ 2}$$

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

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

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

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

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

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

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

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

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

$$\therefore \text{3 }\times {\text{4}}^{\text{4}}\text{ = 3 }\times \text{ 4 }\times \text{ 4 }\times \text{ 4 }\times \text{ 4}$$

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

Hence, the value of $\text{3 }\times {\text{4}}^{\text{4}}$ 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. ${\text{5}}^{\text{2}}\times {\text{3}}^{\text{3}}$

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

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

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

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

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

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

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

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

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

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

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

$$\therefore {\text{3}}^{\text{2}}\times \text{ 1}{\text{0}}^{\text{4}}\text{ = 3 }\times \text{ 3 }\times \text{ 10 }\times \text{ 10 }\times \text{ 10 }\times \text{ 10}$$

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

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

7. Simplify:

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

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

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

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

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

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

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

$$\therefore \left(\text{-3}\right)\times {\left(\text{-2}\right)}^{\text{3}}\text{ = }\left(\text{-3}\right)\times \left(\text{-2}\right)\times \left(\text{-2}\right)\times \left(\text{-2}\right)$$

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

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

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

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

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

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

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

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

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

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

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

Hence, the value of ${\left(\text{-2}\right)}^{\text{3}}\times {\left(\text{-10}\right)}^{\text{3}}$ 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. ${\text{3}}^{\text{2}}\times {\text{3}}^{\text{4}}\times {\text{3}}^{\text{8}}$

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

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

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

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

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

ii. ${\text{6}}^{\text{15}}÷{\text{6}}^{\text{10}}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\Rightarrow {\text{5}}^{\text{6}}÷{\text{5}}^{\text{3}}\text{ = }{\text{5}}^{\text{3}}$

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

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

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

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

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

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

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