Exercise 5B
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 :
$=\left((8)^{5} \times(8)^{6}\right)^{3}$
$=\left((8)^{5+6}\right)^{3}=(8^{11})^3=8^{33}$
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 :
$=\left(\frac{-5}{9}\right)^{4+12+8} = \left(\frac{-5}{9}\right)^{24}$
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 :
$=5^{5 \times 3} \times 5^{4 \times 8}$
$=5^{15} \times 5^{32}=5^{15+32}=5^{47}$
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 :
$=\left(-3 \times \frac{2}{3}\right)^{5}$
$=(-2)^5=-32$
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 :
$=\left(\frac{5}{6}\right)^{6-8}=\left(\frac{5}{6}\right)^{-2}$
$=\left(\frac{6}{5}\right)^2=\frac{6 \times 6}{5 \times 5}=\frac{36}{25}$
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 :
$=(-2)^{15-15}=(-2)^{0}=1$
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
(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
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 :
=$3\times 10^6+4\times 10^5+0 \times 10^4+0 \times 10^3+6 \times 10^2+0\times 10^1+0\times 10^0$
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
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