SChand Composite Mathematics Class 7 Chapter 5 Exponents & Powers Exercise 5B

 Exercise 5B

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Q1 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 1

Fill in the blanks 

(i) $2^{10} \times 2^{8}=2^{\Box}$

Sol :

$2^{10} \times 2^{8}=2^{10+8}$

(ii) $\left(\left(\frac{1}{10}\right)^{3}\right)^{4}=\left(\frac{1}{10}\right)^{\Box}$

Sol :

$\left(\left(\frac{1}{10}\right)^{3}\right)^{4}=\left(\frac{1}{10}\right)^{12}$

(iii) $\left((8)^{5} \times(8)^{6}\right)^{3}=8^{\Box}$

Sol :

$=\left((8)^{5} \times(8)^{6}\right)^{3}$

$=\left((8)^{5+6}\right)^{3}=(8^{11})^3=8^{33}$

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Q2 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 2

Express each of the following product in exponential form :

(i) $7^{3} \times 7^{4}$

Sol :

$=7^{3+4}=7^{7}$

(ii) $(-6) \times(-6)^{5}$

Sol :

$=(-6)^{1+5}=6^{6}$

(iii) $9^{2} \times 9^{18} \times 9^{7}$

Sol :

$=9^{2+18+7}=9^{27}$

(iv) $a^{1} \times a^{2} \times a^{3}$

Sol :

$=a^{1+2+3}=a^{6}$

(v) $\left(\frac{1}{3}\right)^{5} \times\left(\frac{1}{3}\right)^{2}$

Sol :

$=\left(\frac{1}{3}\right)^{5+2}=\left(\frac{1}{3}\right)^{7}$

(vi) $\left(\frac{-5}{9}\right)^{4} \times\left(\frac{-5}{9}\right)^{12} \times\left(\frac{-5}{9}\right)^{8}$

Sol :

$=\left(\frac{-5}{9}\right)^{4+12+8} = \left(\frac{-5}{9}\right)^{24}$

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Q3 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 3

Express each of the following with a single exponents:

(i) $(3^2)^2$

Sol :

$=3^{2+\times 2}=3^{4}$

(ii) $(2^2)^3$

Sol :

$=2^{2\times 3}= 2^6$

(iii) $\left\{(-5)^4 \right\}^2$

Sol :

$=-5^{4 \times 2}=-5^{8}$

(iv) $\left(\left(\frac{1}{5}\right)^{4}\right)^{5}$

Sol :

$=\left(\frac{1}{5}\right)^{4 \times 5}=\left(\frac{1}{5}\right)^{20}$

(v) $\left(\left(\frac{1}{3}\right)^{7}\right)^{3}$

Sol :

$=\left(\frac{1}{3}\right)^{7 \times 3}=\left(\frac{1}{3}\right)^{21}$

(vi) $\left(5^{5}\right)^{3} \times\left(5^{4}\right)^{8}$

Sol :

$=5^{5 \times 3} \times 5^{4 \times 8}$

$=5^{15} \times 5^{32}=5^{15+32}=5^{47}$

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Q4 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 4

Simplify :

(i) $\left(-2 \times 10^{3}\right)^{2}$

Sol :

$=(-2)^{2} \times(10^3)^{2}$

=4000000

(ii) $\left(-\frac{1}{2} \times 5\right)^{2}$

Sol :

$=\left(\frac{-1}{2}\right)^{2} \times\left(\frac{5}{1} \right)^{2}=\left(\frac{25}{4}\right)$

(iii) $\left(\frac{21}{19}\right)^{4} \times\left(\frac{19}{7}\right)^{4}$

Sol :

$=\left(\frac{21}{19} \times \frac{19}{7}\right)^{4}$

$=3^4$=3×3×3×3=81

(iv) $(-3)^{5} \times\left(\frac{2}{3}\right)^{5}$

Sol :

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

$=(-2)^5=-32$

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Q5 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 5

Simplify :

(i) $\frac{5^{4}}{5^{3}}$

Sol :

$=5^{4-3}=5^{1}$

(ii) $\frac{10^{11}}{10^{8}}$

Sol :

$=10^{11-8}=10^{3}$

=1000

(iii) $\frac{(-3)^{10}}{(-3)^{6}}$

Sol :

$=(-3)^{10-6}=(-3)^{4}$

=-3×-3×-3×-3

=81

(iv) $\frac{9^{10}}{9^{13}}$

Sol :

$=9^{10-13}=9^{-3}$

$=\left(\frac{1}{9}\right)^3$

$=\frac{1\times 1 \times 1}{9\times 9 \times 9}=\frac{1}{729}$

(v) $\frac{(-4)^{5}}{(-4)^{8}}$

Sol :

$=(-4)^{5-8}=(-4)^{-3}$

$=\frac{1}{(-4)^3}=\frac{1}{-4 \times -4 \times -4}$

$=\frac{1}{-64}=-\frac{1}{64}$

(vi) $\frac{(-7)^{5}}{(-7)^{5}}$

Sol :

=1

(vii) $\left(\frac{-4}{5}\right)^{12} \div\left(-\frac{4}{5}\right)^{10}$

Sol :

$=\left(\frac{-4}{5}\right)^{12-10}=\left(\frac{-4}{5}\right)^{2}$

$=\frac{-4 \times -4}{5 \times 5}=\frac{16}{25}$

(viii) $\left(\frac{5}{6}\right)^{6} \div\left(\frac{5}{6}\right)^{8}$

Sol :

$=\left(\frac{5}{6}\right)^{6-8}=\left(\frac{5}{6}\right)^{-2}$

$=\left(\frac{6}{5}\right)^2=\frac{6 \times 6}{5 \times 5}=\frac{36}{25}$

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Q6 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 6

Evaluate :

(i) $(419)^0$

Sol :

=1

(ii) $(-17)^{0}$

Sol :

=1

(iii) $(100000)^0$

Sol :

=1

(iv) $\left(-\frac{5}{7}\right)^{0}$

Sol :

=1

(v) $7^{0}+8^{0}+9^{0}$

Sol :

=1+1+1=3

(vi) $\left(29^{0}-23^{0}\right) \times 16^{0}$

Sol :

=(1-1)1=0

(vii) $(-2)^{3 \times 5-4-11}$

Sol :

$=(-2)^{15-15}=(-2)^{0}=1$

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Q7 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 7

Find the value of x in each case :

(i) $(2)^{x+2}=256$

Sol :

$=256=16^2=(4^2)^2=\{(2^2)^2\}^2=2^8$

$(2)^{x+2}=2^8$

On comparing 

x+2=8

x=8-2=6

(ii) $x^{3}=729$

Sol :

$x^{3}=9^3$

On comparing 

x=9

(iii) $(-6)^{5} \times(-6)^{3-m}=(-6)^{3}$

Sol :

$(-6)^{5+3-m} =(-6)^{3}$

On comparing 

5+3-m=3

8=3+m

8-3=m

5=m

or m=5

(v) $49 \times(-7)^{m}=-343$

Sol :

$(-7)^{m}=\frac{-343}{49}$

-7m=-7

$m=\frac{-7}{-7}=1$

(vi) $\left(-\frac{8}{9}\right)^{15} \div\left(-\frac{8}{9}\right)^{x}=\left(-\frac{8}{9}\right)^{2}$

Sol :

$\left(-\frac{8}{9}\right)^{15-x}=\left(-\frac{8}{9}\right)^{2}$

15-x=2

15-2=x

13=x or x=13

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Q8 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 8

Rename the number in expanded form using exponents:

(i) 78093

Sol :

$=7 \times 10^{4}+8 \times 10^{3}+0 \times 10^{2}+9 \times 10^{1}+3 \times 10^{0}$

(ii) 850000

Sol :

$=8 \times 10^{5}+5 \times 10^{4}$

(iii) 154034

Sol :

$=1\times 10^{5}+5 \times 10^{4}+4 \times 10^{3}+3 \times 10^{1}+4 \times 10^{0}$

(iv) 34,00,600

Sol :

=$3\times 10^6+4\times 10^5+0 \times 10^4+0 \times 10^3+6 \times 10^2+0\times 10^1+0\times 10^0$

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Q9 | Ex-5B |Class 7 | Exponents & Powers | S.Chand |Composite Mathematics |Chapter 5 |myhelper

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Question 9

Simplify :

(i) $\frac{\left(3^{3}\right)^{2} \times 5^{2}}{9^{2} \times 5}$

Sol :

$=\frac{\left(3^{2}\right)^{3} \times 5^{2}}{9^{2} \times 5}=9^{3-2} \times 5^{2-1}$

=9×5=45

(ii) $\frac{\left(\frac{3}{5}\right)^{3} \times\left(\frac{1}{7}\right)^{3}}{\left(\frac{3}{5}\right)^{2} \times\left(\frac{1}{7}\right)^{4}}$

Sol :

$=\frac{\left(\frac{3}{5}\right)}{\left(\frac{1}{7}\right)}$

$=\frac{3}{5} \times \frac{7}{1}=\frac{21}{5}$

(iii) $\frac{12^{4} \times 9^{3} \times 4}{6^{3} \times 8^{2} \times 27}$

Sol :

$=\frac{6^{4} \times 2^{4} \times 3^{6} \times 4}{6^3 \times 4^{2} \times 2^{2} \times 3^{3}}$

$=\frac{6^{1} \times 2^{2} \times 3^3}{4}=162$

(iv) $\frac{3^5 \times 10^{5} \times 25}{5^{7} \times 6^5}$

Sol :

$=\frac{3^{5} \times(2 \times 5)^{5} \times 5^{2}}{5^{7} \times(2 \times 3)^{5}}$

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

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

=1

(v) $\left[\left\{\left(-\frac{1}{3}\right)^{2}\right)^{-2}\right]^{-1}$

Sol :

$=\left(-\frac{1}{3}\right)^4=\frac{1}{81}$

(vi) $\left(2^{-1} \div 5^{-1}\right) \times\left(\frac{-5}{8}\right)^{-1}$

Sol :

$=\left(\frac{1}{2} \div \frac{1}{5}\right) \times \frac{8}{-5}$

$=\frac{1}{2} \times \frac{5}{1} \times \frac{8}{-5}$

=-4