Laws of exponents class 7

Contents

What is an exponent?

Exponents are used to express large numbers in shorter form to make them easy to read, understand, compare and operate upon. The base is the number that is being repeated as a factor in the multiplication. For example,

$\large 4 \times 4\times 4 =4^{3}$

The exponent tells you how many times the base is repeated as a factor in the multiplication . Here exponent is 3 .

Exponent rules

Laws of exponents class 7

Laws of exponents Class 7 questions

Laws of exponents class 7 (solved examples)

Simplify the following using laws of exponents class 7

Example 1. Find the value of $\inline \large \left ( 4^{0}-3^{0} \right ) \times 7^{0}$ .

Solution. Since for any non- zero integer $a, \, a^{0}=1$ , so $\inline \large \left ( 4^{0}-3^{0} \right ) \times 7^{0}$ = (1-1) x 1 = 0 x 1 =0.

Hence $\inline \large \left ( 4^{0}-3^{0} \right ) \times 7^{0}$ = 0 .

Example 2 . Evaluate $\inline \large \left ( 3^{55} \times 3^{60 }\right )- \left ( 3^{97} \times 3^{18 }\right )$

Solution. Using the rule $a^{m}\times a^{n} =a^{m+n}$ , we get $\inline \large \left ( 3^{55} \times 3^{60 }\right )- \left ( 3^{97} \times 3^{18 }\right )$ = $\inline \large 3^{55+60}-3^{97+18}$

= $\large 3^{115}-3^{115}$

Example 3. Express 528 in exponential notation.

Solution. 528= 2 x 2 x 2 x 2 x 3 x 11

= $\large 2^{4}\times 3 \times 11$

exponential form of 528

Example 4. Find x such that $\left ( \frac{1}{3} \right )^{5}\times \left ( \frac{1}{3} \right )^{15}=\left ( \frac{1}{3} \right )^{4x}$ .

Solution. Using the law of exponents $a^{m}\times a^{n} =a^{m+n}$ , we get

$\left ( \frac{1}{3} \right )^{5+15}=\left ( \frac{1}{3} \right )^{4x}$

$\left ( \frac{1}{3} \right )^{20}=\left ( \frac{1}{3} \right )^{4x}$

On both the sides , powers have the same base , so their exponents must be equal. Therefore 4x= 20 or x=5.

Hence the value of x is 5.

Example 5. Solve $\mathbf{\frac{10^{22}+10^{20}}{10^{20}}}$ .

Solution. $\frac{10^{22}+10^{20}}{10^{20}}$ = $\frac{10^{22}}{10^{20}} +1$ = $10^{22-20}+1$ ( Using the rule $\frac{a^{m}}{a^{n}}=a^{m-n}$ )

= $10^{2} +1=100+1=101$

Example 6. Find the value of $\mathbf{k}$ if, $\mathbf{3^{1998}-3^{1997}-3^{1996}+3^{1995} =k . 3^{1995}}$

Solution. $\large 3^{1998}-3^{1997}-3^{1996}+3^{1995}= k . 3^{1995}\Rightarrow 3^{1995}\left ( \frac{3^{1998}}{3^{1995}} - \frac{3^{1997}}{3^{1995}}-\frac{3^{1996}}{3^{1995}}+1\right )=$ $k.3^{1995}$

$\large \Rightarrow 3^{(1998-1995)}-3^{(1997-1995)}-3^{(1996-1995)}+1 =k$

$\large \Rightarrow k=3^{3}-3^{2}-3+1= 27-9-3+1 = 16$

Example 7. By what number should we multiply $4^{4}$ so that the product may be equal to $4^{8}$ ?

Solution. Let $4^{4}$ be multiplied by $x$ so that the product may be equal to $4^{8}$ .
According to question, $\large 4^{4} \times x= 4^{8}$

or $x=\frac{4^{8}}{4^{4}}$

$\Rightarrow x=4^{(8-4))}=4^{4}$ =256 ( Using the rule $\frac{a^{m}}{a^{n}}=a^{m-n}$ )

Therefore , $4^{4}$ should be multiplied by 256 so that the product is equal to $4^{8}$ .

Example 8. Solve $\large \frac{2^{4}}{\left ( 7^{0}+3^{0} \right )^{3}}$

Solution. $\frac{2^{4}}{\left ( 7^{0}+3^{0} \right )^{3}}$ = $\frac{16}{\left ( 1+1 \right )^{3}}$ $=\frac{16}{2^{3}}$ = $\frac{16}{8}=2$

Example 9. Find m, so that $\left ( \frac{2}{11} \right )^{3}\times \left ( \frac{2}{11} \right )^{6}=\left ( \frac{2}{11} \right )^{2m-1}$ .

Solution. We have, $\left ( \frac{2}{11} \right )^{3}\times \left ( \frac{2}{11} \right )^{6}=\left ( \frac{2}{11} \right )^{2m-1}$

$\Rightarrow \left ( \frac{2}{11} \right )^{3+6}=\left ( \frac{2}{11} \right )^{2m-1}$ ( Using the law of exponents $a^{m}\times a^{n} =a^{m+n}$ )

$\Rightarrow \left ( \frac{2}{11} \right )^{9}=\left ( \frac{2}{11} \right )^{2m-1}$

On comparing both the sides, we get 9= 2m-1

$\Rightarrow 2m=9+1=10$

$\Rightarrow m=\frac{10}{2}=5$ . Hence $m=5$

Example 10. If $\frac{p}{q}=\left ( \frac{3}{2} \right )^{2}$ ÷ $\left ( \frac{2}{7} \right )^{0}$ , then find the value of $\left ( \frac{p}{q} \right )^{3}$ .

Solution. For any non- zero integer $a ,\, \, a^{0}=1$ .Hence $\frac{p}{q}=\left ( \frac{3}{2} \right )^{2}$ ÷ 1

$\Rightarrow$ $\frac{p}{q}=\frac{3^{2}}{2^{2}}=\frac{9}{4}$

On cubing both the sides, we get $\left ( \frac{p}{q} \right )^{3} =\frac{9^{3}}{4^{3}}=\frac{9\times 9\times 9}{4\times 4\times 4}=\frac{729}{64}$ .

Powers and Exponents worksheet pdf

Exponents and Powers Class 7 Extra Questions

A. Fill in the blanks:

(i) $(-1)^{even \, \, number} =$ ……….

(ii) $(-1)^{odd \, number }= ......$

(iii) $\left ( \frac{-2}{3}\right )^{5}$ =………

(iv) $(-8)^{3} \times (-8)^{5}$ =…………..

(v) For any two non-zero integers $x$ any $y$ , $x^{5}$ ÷ $y^{5}$ is equal to……………..

(vi) ( $5^{9}$ ÷ $5^{8})^{2}$ =……….

(vii) $\left ( -1 \right )^{12} + \left ( -1 \right )^{143}+\left ( -1 \right )^{13}$ = …………….

(viii) $8 \times( \frac{2}{9})^{0}$ =……………..

(ix) The prime factorisation of 216 in exponential form is = ………………

(x) 2401 as a power of 7 = ……………..

B. State whether the following statements are True /False.

(i) $\frac{a^{6}}{b^{2}}=\frac{a+a+a+a+a+a}{b+b}$

(ii) $\left [ \left ( -3 \right )^{2} \right ]^{5} =\left ( -3 \right )^{10}$

(iii) For a non-zero rational number $x , \, \, x^{5}$ ÷ $x^{3}=x^{2}$

(iv) $1^{0}+2^{0}+3^{0}+4^{0}+5^{0} =4$

(v) $3^{2}> 2^{3}$

(vi) $x^{0}\times x^{0}=x^{0}$ ÷ $x^{0}$ is true for all non-zero values of $x$ .

(vii) 1° x 0¹ =1

(viii) $\left ( \frac{-2}{5} \right )^{50}=\frac{-2^{50}}{-5^{50}}$

(ix) $\left ( 5+5 \right )^{10}=5^{10}+5^{10}$

(x) x^m + x^m = x^2m, where x is a non-zero rational number and m is a positive integer.

C. (i) By what number should we multiply $3^{3}$ so that the product may be equal to $3^{6}$ ?

(ii) Find $x$ so that $\left ( \frac{3}{2} \right )^{5} \times \left ( \frac{3}{2} \right )^{11}=\left ( \frac{3}{2} \right )^{2x}$

(iii) If a=2 and b=3 then find the value of $a^{b}+b^{a}$ .

(iv) If $\left ( -3 \right )^{8}$ ÷ $\left ( -3 \right )^{5}$ = $\left ( -3 \right )^{x}$ then find the value of $x$ .

(v) For non-zero numbers $a$ and $b$ , $\left ( \frac{a}{b} \right )^{m}$ ÷ $\left ( \frac{a}{b} \right )^{n}$ , where $m>n$ , is equal to ……………….

(vi) Express each of the following in single exponential form,
(a) 3³ x 4³
(b) 2⁴ x 4²
(c) 6² x 8²
(d) (- 6)⁵ x (- 6)
(e) (- 3)³ x (- 10)³
(f) (- 11)² x (- 5)²

Ans. A. (i) 1 (ii) -1 (iii) $-\frac{32}{27}$ (iv) $\left ( -8 \right )^{8}$ (v) $\left ( \frac{x}{y} \right )^{5}$ (vi) $5^{2}$ (vii) -1 (viii) 8 (ix) $2^{3} \times 3^{3}$ (x) $7^{4}$

B. (i) False (ii) True (iii) True (iv) False (v) True (vi) True (vii) False (viii) True (ix) False (x) False

C. (i) 27 (ii) x=8 (iii) 17 (iv) 3 (v) $\left ( \frac{a}{b} \right )^{m-n}$ (vi) a) $12^{3}$ b) $2^{8}$ c) $\left ( 48 \right )^{2}$ d) $6^{6}$ e) $\left ( 30 \right )^{3}$ f) $55^{2}$