Important Questions for CBSE Class 7 Maths Chapter 13 - Exponents and Powers

Very Short Answer Questions                                                                  1 Mark

1. Find $\mathbf{{2^8}}$.

Ans: Given: ${2^8}$

We need to find the value of the given exponent.

We can rewrite ${2^8}$ to find its value as

${2^8} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

${2^8} = 256$

2. Express the following in exponential form $\mathbf{2 \times 2 \times a \times a}$.

Ans: Given: $2 \times 2 \times a \times a$

We need to write the given expression as an exponential form.

A number can be written in its exponential form if we raise the power of the number by the exponent.

Therefore, exponential form of $2 \times 2 \times a \times a$ is

$2 \times 2 \times a \times a $

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

$ = 4{a^2} $

3. Find $\mathbf{{( - 4)^3}}$.

Ans: Given: ${( - 4)^3}$

We need to find the value of a given exponent.

We can rewrite ${( - 4)^3}$ to find its value as

${( - 4)^3} =  - 4 \times  - 4 \times  - 4 $

${( - 4)^3} =  - 64 $

4. $\mathbf{{a^m} \times {a^n}}$=_______?

Ans: Given: ${a^m} \times {a^n}$

We need to fill in the blanks.

Therefore, ${a^m} \times {a^n}$$ = \underline {{a^{m + n}}} $

5. $\mathbf{{a^0} = \_\_\_\_\_?}$

Ans: Given: ${a^0}$

We need to find the value of a given expression.

We know that if $0$ is the power of any number then the value of the number is always $1.$

Therefore, ${a^0} = \underline {1.} $

Short Answer Questions                                                                          2 Mark

6. Express 16807 in exponential form.

Ans: Given: $16807$

We need to express the given number in exponential form.

Exponential form is a way to represent a number in repeated multiplications of the same number.

So, we can write $16807$ as
$16807 = 7 \times 7 \times 7 \times 7 \times 7 $

$16807 = {7^5} $

7. Identify which is greater $\mathbf{{2^7}{\text{ or }}{7^2}}$.

Ans: Given: exponents ${2^7},{7^2}$

We need to find which exponent is greater. 

We will find the value of each exponent and then compare it.

We can write the exponents as

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

${2^7} = 128 $

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

${7^2} = 49 $

Clearly, we can see that 

${2^7} > {7^2}$

8. Simplify $\mathbf{{7^3} \times {2^5}}$.

Ans: Given: ${7^3} \times {2^5}$

We need to simplify the given exponential expression.

We can simplify the given expression as 

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

${7^3} = 343 \times 32 $

${7^3} = 10976 $

9. Write 1024 as a power of 2.

Ans: Given: $1024$

We need to write the given expression as power of $2$

Break $1024$ in factors of 2 and write as exponents.

Therefore, $1024$ as power of $2$ will be written as

$1024 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 $

$\Rightarrow 1024 = {2^{10}} $

10. Using laws, find the value of $\mathbf{\left( {{3^{15}} \div {3^{10}}} \right) \times {3^2}}$.

Ans: Given: $\left( {{3^{15}} \div {3^{10}}} \right) \times {3^2}$

We need to find the value of a given expression using laws.

We know that

$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $

${a^m} \times {a^n} = {a^{m + n}} $

Using these laws, the value of $\left( {{3^{15}} \div {3^{10}}} \right) \times {3^2}$ will be

$= \left( {{3^{15}} \div {3^{10}}} \right) \times {3^2} $

$= \dfrac{{{3^{15}}}}{{{3^{10}}}} \times {3^2} $

$= {3^{15 - 10}} \times {3^2} $

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

$= {3^{5 + 2}} $

$= {3^7} $

 $= 2187 $

11. Find $\mathbf{8 \times {10^5} + 0 \times {10^4} + 3 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 5 \times {10^0}}$.

Ans: Given: $8 \times {10^5} + 0 \times {10^4} + 3 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 5 \times {10^0}$

We need to find the value of the given expression.

We will solve the given exponents and then add them.

Therefore, the value of $8 \times {10^5} + 0 \times {10^4} + 3 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 5 \times {10^0}$ will be

$= 8 \times 100000 + 0000 + 3 \times 1000 + 2 \times 100 + 00 + 5 \times 1 $

$= 800000 + 0 + 3000 + 200 + 0 + 5 $

$= 803205 $ 

12. Say True or False and Justify.

1. $\mathbf{{5^2} > {4^3}}$

Ans: Given: ${5^2} > {4^3}$

We need to find if the given expression is true or false.

We will solve the exponents and then compare them.

${5^2} = 25 $

${4^3} = 64 $

$25 < 64 $

$\Rightarrow {5^2} < {4^3} $

Therefore, the expression is False.

2. $\mathbf{{5^0} = {343^0}}$

Ans: Given: ${5^0} = {343^0}$

We need to find if the given expression is true or false.

We will solve the exponents and then compare them.

${5^0} = 1 $

${343^0} = 1 $

$\therefore {5^0} = {343^0} $

Therefore, the expression is true.

13. Find the value of $\mathbf{\left( {{3^0} + {2^0}} \right) \times {5^1}}$.

Ans: Given: $\left( {{3^0} + {2^0}} \right) \times {5^1}$

We need to find the value of a given expression.

We know that ${a^0} = 1$

Therefore, the value of $\left( {{3^0} + {2^0}} \right) \times {5^1}$ will be

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

$= (1 + 1) \times 5 $

$= 2 \times 5 $

$= 10 $

14. Find $\mathbf{\left( {\dfrac{{{a^6}}}{{{a^4}}}} \right) \times {a^{ - 2}} \times {a^0}}$.

Ans: Given: $\left( {\dfrac{{{a^6}}}{{{a^4}}}} \right) \times {a^{ - 2}} \times {a^0}$

We need to find the value of the given expression.

We know that 

$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $

${a^m} \times {a^n} = {a^{m + n}} $

${a^0} = 1 $ 

Therefore, $\left( {\dfrac{{{a^6}}}{{{a^4}}}} \right) \times {a^{ - 2}} \times {a^0}$ will be

$= \left( {{a^{6 - 4}}} \right) \times {a^{ - 2}} \times {a^0} $

$= {a^2} \times {a^{ - 2}} \times 1 $

$= {a^{2 + ( - 2)}} $

$= {a^0} $

$= 1 $ 

15. Find $\mathbf{{27^p} \div {27^2}}$.

Ans: Given: ${27^p} \div {27^2}$

We need to find the given expression.

We know that 

$\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Therefore, ${27^p} \div {27^2}$ will be

$= {\left( {{3^3}} \right)^p} \div {\left( {{3^3}} \right)^2} $

$= \dfrac{{{3^{3p}}}}{{{3^6}}} $

$= {3^{3p - 6}} $

$= {3^{3(p - 2)}} $ 

Short Answer Questions                                                                          2 Mark

16. Express each of the following as product of prime factor

1. $\mathbf{702}$

Ans: We need to express the given expression as product of prime factor

Exponential form is a way to represent a number in repeated multiplications of the same number.

Therefore, $702$ can be written as a product of prime factors as

$702 = 2 \times 3 \times 3 \times 3 \times 13 $

$= {2^1} \times {3^3} \times {13^1} $

2. $\mathbf{33275}$

Ans: Given: $33275$

We need to express the given expression as a product of prime factors.

Exponential form is a way to represent a number in repeated multiplications of the same number.

Therefore, $33275$ can be written as a product of prime factors as

$33275 = 5 \times 5 \times 11 \times 11 \times 11 $

$= {5^2} \times {11^3} $

17. Using the laws find

1. $\mathbf{\left( {{{\left( {{3^3}} \right)}^2} \times {3^2}} \right) \div {3^7}}$

Ans: Given:  $\left( {{{\left( {{3^3}} \right)}^2} \times {3^2}} \right) \div {3^7}$

We need to find the value of a given expression using laws.

We know that

$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $

${a^m} \times {a^n} = {a^{m + n}} $ 

Therefore, the value of $\left( {{{\left( {{3^3}} \right)}^2} \times {3^2}} \right) \div {3^7}$ will be

$= \left( {{3^6} \times {3^2}} \right) \div {3^7} $

$= \left( {{3^{6 + 2}}} \right) \div {3^7} $

$= {3^8} \div {3^7} $

$= {3^{8 - 7}} $

$= {3^1} $

$= 3 $ 

2. $\mathbf{\dfrac{{{3^6}{a^8}{b^4}}}{{{3^2}{a^2}{b^3}}}}$

Ans: Given:  $\dfrac{{{3^6}{a^8}{b^4}}}{{{3^2}{a^2}{b^3}}}$

We need to find the value of a given expression using laws.

We know that

$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $

${a^m} \times {a^n} = {a^{m + n}} $
Therefore, the value of $\dfrac{{{3^6}{a^8}{b^4}}}{{{3^2}{a^2}{b^3}}}$ will be

$= \dfrac{{{3^6}{a^8}{b^4}}}{{{3^2}{a^2}{b^3}}} $

$= {3^{6 - 2}} \times {a^{8 - 2}} \times {b^{4 - 3}} $

$= {3^4} \times {a^6} \times {b^1} $

$= 81{a^6}{b^1} $ 

18. Express each of the following as product of prime factors

1. $\mathbf{729 \times 625}$

Ans: We need to express the given expression as product of prime factor

Exponential form is a way to represent a number in repeated multiplications of the same number.

Therefore, $729 \times 625$ can be written as a product of prime factors as

$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 $ 

$= {3^6} $ 

$625 = 5 \times 5 \times 5 \times 5 $

$= {5^4} $

$\therefore 729 \times 625 = {3^6} \times {5^4} $

2. $\mathbf{1024 \times 216}$

Ans: Given: $1024 \times 216$

We need to express the given expression as a product of prime factors.

Exponential form is a way to represent a number in repeated multiplications of the same number.

Therefore, $1024 \times 216$ can be written as a product of prime factors as

\[\begin{align} & 1024=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2={{2}^{10}} \\ & 216=2\times 2\times 2\times 3\times 3\times 3={{2}^{3}}\times {{3}^{3}} \\  & \therefore 1024\times 216={{2}^{10}}\times \ {{2}^{3}}\times {{3}^{3}} \\  & \,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ={{2}^{10+3}}\times {{3}^{3}} \\  & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ={{2}^{13}}\times {{3}^{3}} \\ \end{align}\]

19. Express the following as standard form

1. $\mathbf{3,68,878}$

Ans: Given: $3,68,878$

We need to express the given number as a standard form.

We will write the given numbers as a multiple of power of $10.$

Therefore, the standard form of $3,68,878$ will be

$= 3.68878 \times 100000 $

$= 3.68878 \times {10^5} $ 

2. $\mathbf{4,78,25,00,000}$

Ans: Given: $4,78,25,00,000$

We need to express the given number as a standard form.

We will write the given numbers as a multiple of power of $10.$

Therefore, the standard form of $4,78,25,00,000$ will be

$= 4.7825 \times 1000000000 $

$= 4.7825 \times {10^9} $ 

3. $\mathbf{5706.983}$

Ans: Given: $5706.983$

We need to express the given number as a standard form.

We will write the given numbers as a multiple of power of $10.$

Therefore, the standard form of $5706.983$ will be

$= 5.706983 \times 1000 $

$= 5.706983 \times {10^3} $ 

20. Find the numbers

1. $\mathbf{8 \times {10^5} + 2 \times {10^2} + 4 \times {10^1}}$

Ans: Given: $8 \times {10^5} + 2 \times {10^2} + 4 \times {10^1}$

We need to solve the given expression.

We will solve the exponents and then add them.

Therefore, $8 \times {10^5} + 2 \times {10^2} + 4 \times {10^1}$ will be 

$= 8 \times 100000 + 0 \times 0 + 2 \times 100 + 4 \times 10 $

$= 800000 + 0 + 0 + 200 + 40 $

$= 8,00,240 $ 

2. $\mathbf{4 \times {10^4} + 6 \times {10^3} + 2 \times {10^2} + 1 \times {10^0}}$

Ans: Given: $4 \times {10^4} + 6 \times {10^3} + 2 \times {10^2} + 1 \times {10^0}$

We need to solve the given expression.

We will solve the exponents and then add them.

Therefore, $4 \times {10^4} + 6 \times {10^3} + 2 \times {10^2} + 1 \times {10^0}$ will be 

$= 4 \times 10000 + 6 \times 1000 + 2 \times 100 + 1 \times 1 $

$= 40000 + 6000 + 200 + 1 $

$= 46,201 $ 

3. $\mathbf{5 \times {10^4} + 7 \times {10^2} + 5 \times {10^0}}$

Ans: Given: $5 \times {10^4} + 7 \times {10^2} + 5 \times {10^0}$

We need to solve the given expression.

We will solve the exponents and then add them.

Therefore, $5 \times {10^4} + 7 \times {10^2} + 5 \times {10^0}$ will be 

$= 5 \times 10000 + 7 \times 100 + 5 \times 1 $

$= 50000 + 700 + 5 $

$= 50,705 $ 

Definition of Exponent

The exponent tells us how many times a number should be multiplied by itself to get the desired result. Thus any number (suppose a) raised to power ‘n’ can be expressed as:

a\[^{n}\] = a x a x a x a x a x a…. x a(n times)

Here a can be any number and n is the natural number.

a\[^{n}\] is also called the n\[^{th}\] power of a.

In this ‘a’ is the base and ‘n’ is the exponent or index or power.

 ‘a’ is multiplied ‘n’ times, It is a method of repeated multiplication.

a\[^{m}\] . a\[^{n}\] = a\[^{(m+n)}\]

(a\[^{m}\])\[^{n}\] = a\[^{mn}\]

(ab)\[^{n}\] = a\[^{n}\]b\[^{n}\]

(\[\frac{a}{b}\])\[^{n}\] = \[\frac{a^{n}}{b^{n}}\]

\[\frac{a^{m}}{a^{n}}\] = a\[^{m-n}\]

\[\frac{a^{m}}{a^{n}}\] = \[\frac{1}{a^{n-m}}\]

a\[^{0}\] = 1

Multiplying Powers With the Same Base

When the bases are the same then we add the exponents. 

a\[^{m}\] x a\[^{n}\] = a\[^{(m+n)}\]

In a similar way, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then

a\[^{m}\] x a\[^{n}\] = a\[^{(m+n)}\]

Similarly (\[\frac{a}{b}\])\[^{m}\] x (\[\frac{a}{b}\])\[^{n}\] = (\[\frac{a}{b}\])\[^{m+n}\]   

Note:

•  Exponents can be added only when the bases are the same. 

•  Exponents cannot be added if the bases are not the same.

Dividing Powers with the Same Base

In division, if the bases are the same then we need to subtract the exponents.

a\[^{m}\]  ÷  a\[^{n}\] =  \[\frac{a^{m}}{a^{n}}\] = a\[^{m-n}\]   

Where m and n are whole number and m<n;

We can generalize that if a is a non-zero integer or q non-zero rational number and m and n are positive integers, such that m<n;

a\[^{m}\]  ÷  a\[^{n}\] = a\[^{m-n}\] if m<n, then a\[^{m}\]  ÷  a\[^{n}\] = \[\frac{1}{a^{(n-m)}}\]    

Similarly, (\[\frac{a}{b}\])\[^{m}\] ÷ (\[\frac{a}{b}\])\[^{n}\] = (\[\frac{a}{b}\])\[^{m-n}\]  

Power of a Power

In the power of a power you need multiply the powers

In general, for any non-integer a (a\[^{m}\])\[^{n}\] = a\[^{m \times n}\] = a\[^{mn}\]

Multiplying Power with the Same Exponent

In general, for any non-zero integer a,b

a\[^{m}\] x b\[^{m}\] = (a x b)\[^{m}\] = (ab)\[^{m}\]

Negative Exponents

If the exponent is negative we need to change it into a positive exponent by writing the same in the denominator and 1 in the numerator.

If ‘a’ is a non-zero integer or a non-zero rational number and m is a positive integer, then a\[^{-m}\] is the reciprocal of, i.e., 

a\[^{-m}\] = \[\frac{1}{a^{m}}\], if we take a as p/q then 

(\[\frac{p}{q}\])\[^{-m}\] = \[\frac{1}{(\frac{p}{q})^{m}}\] = (\[\frac{q}{p}\])\[^{m}\]

Also, \[\frac{1}{a^{-m}}\] = a\[^{m}\]  

Similarly, (\[\frac{a}{b}\])\[^{-m}\] = (\[\frac{b}{a}\])\[^{m}\], where n is a positive integer

Power with Exponent Zero

If the exponent is 0 then you get the result 1 whatever the base is.

If ‘a’ is a non-zero integer or a non-zero rational number then, 

a\[^{0}\] = 1

Similarly, (\[\frac{a}{b}\])\[^{0}\] = 1

Fractional Exponent

In fractional exponent, we observe that the exponent is in fraction form.

a\[^{\frac{1}{n}}\], where a is called the base and 1/n is called an exponent or power. It is denoted as \[\sqrt[n]{a}\] is called as the nth root of a.

Some Rules to Remember While Calculating the Power of a Number

Rule 1: Any number to the zero power is equal to 1.

Rule 2: Any number to the first power is equal to the number itself.

Rule 3: If the base to which we are calculating power is negative, then odd power results in negative values and even power results in positive values.

For example:

(-4)\[^{3}\]= -64

4\[^{2}\] = 16

Rule 4: The exponent comes before the multiplication in the order of operations. We can add in the parentheses if it helps us to solve the question.

For example:

(2 x 3)\[^{2}\] = 6\[^{2}\] = 36  

2 x 3\[^{2}\] = 2 x 9 = 18

The sequence formed by powers of a number (exponent starting from 0 and having integral values) is a geometric progression with a first-term equal to 1 and common ratio being equal to the base.

Look at the pattern below:

2\[^{0}\] = 1

2\[^{1}\] = 2

2\[^{2}\] = 2 x 2 = 4

2\[^{3}\] = 2 x 2 x 2 = 8

2\[^{4}\] = 2 x 2 x 2 x 2 = 16

2\[^{5}\] = 2 x 2 x 2 x 2 x 2 = 32

A common mistake is to multiply the base and exponent together, which is not the correct way to calculate the power of a number.

For example: 

4\[^{3}\] = 4 x 3 = 12 (Wrong)

4\[^{3}\] = 4 x 4 x 4 = 64 (Right)

Why are the Important Questions from Vedantu Useful for Class 7 Maths Chapter 13 - Exponents and Powers?

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Conclusion

Exponents play a crucial role in algebra, simplifying repeated multiplication. The exponent indicates how many times a number multiplies itself. It's important to note that any number to the power of 0 equals 1. When expressing with exponents, attention to negative signs and parentheses is vital. Exponents come in four types: positive, negative, zero, and rational/fraction. To delve deeper into this chapter, access the Class 7 Maths Chapter 13 extra questions PDF on Vedantu’s website or app. This resource enhances understanding and consolidates knowledge about exponents in a user-friendly format.