SChand CLASS 9 Chapter 6 Indices/Exponent Exercise 6(B)

 Exercise 6(B)

 Question 1

Evaluate:

(i) 5°

(ii) $2^{-3}$

(iii) $10^{-4}$

(iv) $(- 2)^{-2}$

(v) $\left(\frac{2}{3}\right)^{\circ}$

(vi) $\left(\frac{3}{4}\right)^{-3}$

(vii) $\left(\frac{−1}{2}\right)^{-2}$

(viii) $\frac{2^{−2}}{5^{−2}}$

(ix) $\frac{3x^0−1}{3x^0+1}$

Sol :

(i) $5^{\circ}$

= $5^{\circ}$=1   $\left(\because x^{0}=1\right)$

(ii) $2^{-3}$

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

$8^{-1}$

$\frac{1}{8}$

(iii) $10^{-4}$

$10^{-1} \times 10^{-1} \times 10^{-1} \times 10^{-1}$

$10000^{-1}$

$\frac{1}{10000}$

(iv) $(-2)^{-2}$

$\frac{1}{(-2)^{2}}$

$\frac{1}{-2 x-2}$

$\frac{1}{4}$

(v) $\left(\frac{2}{3}\right)^{0}$

=1

(vi) $\left(\frac{3}{4}\right)^{-3}$

$\left(\frac{4}{3}\right)^{3} \Rightarrow \frac{4 \times 4 \times 4}{3 \times 3 \times 3} \Rightarrow \frac{64}{27}$

(vii) $\left(-\frac{1}{2}\right)^{-2}$

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

$(-2)^{2}$

$-2 x-2 \Rightarrow 4$

(viii) $\frac{2^{-2}}{5-2}$

$\frac{5^{2}}{2^{2}}$

$\frac{5 \times 5}{2 \times 2} \Rightarrow \frac{25}{4}$

(ix)  $\frac{3 x^{0}-1}{3 x^{0}+1}$

$\frac{3 x 1-1}{3 \times 1+1}$

$\frac{3-1}{3+1}$

$\frac{2}{4} \Rightarrow \frac{1}{2}$

Question  2

Evaluate:

(i) $4\frac{1}{2}$

(ii) $8\frac{1}{3}$

(iii) $16\frac{1}{4}$

(iv) $-27\frac{1}{3}$

(v) $32\frac{3}{5}$

(vi) $(125)^\frac{−2}{3}$

(vii) $\left(\frac{8}{125}\right)^\frac{1}{3}$

(viii) $\left(\frac{1}{216}\right)^{\frac{−2}{3}}$

(ix) $-(- 27)^\frac{−4}{3}$

(x) $\left(\frac{27}{8}\right)^\frac{−2}{3}$

(xi) $(0.0625)^{\frac{3}{4}}$

(xii) $\left(12\frac{19}{27}\right)^\frac{1}{3}$

Sol:

(i) $4 \frac{1}{2}$

$(2 \times 2)^{\frac{1}{4}}$

$\left(2^{2}\right)^{\frac{1}{2}}$

$2\left(2 \times \frac{1}{2}\right)$

$2^{1} \Rightarrow 2$

(ii)$8 \frac{1}{3}$

$(2 \times 2 \times 2)^{\frac{1}{3}}$

$2^{3 \times \frac{1}{3}}$

=21=2

(iii) $16^{\frac{1}{4}}$

$(2 \times 2 \times 2 . \times 2)^{\frac{1}{4}}$

$\left(2^{4}\right)^{\frac{1}{4}}$

=21=2

(iv)  $(-27)^{\frac{1}{3}}$

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

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

$(-3)^{1}$

$(-3)^{1} \Rightarrow-3$

(v) $32^{\frac{3}{5}}$

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

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

$2 \times 2 \times 2=8$

(vi) $(125)^{-3}$

$(5 \times 5 \times 5)^{-2}$

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

$5^{3 \times \frac{-2}{3}}$

$\Rightarrow 5^{-2}$

$\frac{1}{52} \Rightarrow \frac{1}{5 \times 5} \Rightarrow \frac{1}{25}$

(vii) $\left(\frac{8}{125}\right)^{1 / 3}$

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

$\frac{2^{3 \times \frac{1}{3}}}{5^{3 \times \frac{1}{3}}} \Rightarrow \frac{2}{5}$

(viii)$\left(\frac{1}{216}\right)^{-\frac{2}{3}}$

$\left(\frac{1}{6 \times 6 \times 6}\right)^{-\frac{2}{3}}$

=$\left(\frac{1}{(6)^{3}}\right)^{\frac{-2}{3}}$

=$\frac{(1)^{\frac{-2}{3}}}{\left(6^{3}\right)^{\frac{-2}{3}}}$

=$\frac{1}{(6)^{-2}}$

=$(6)^{2}$

=36

(ix) $-(-27)^{-4\3}$

=$\Rightarrow-[(-3)(-3)(-3)]^{-\frac{4}{3}}$

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

=$\left[(-3)^{3 x}\left(-\frac{-4}{3}\right)\right]$

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

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

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

= $\frac{-1}{81}$

(x) $\left[\frac{27}{8}\right]^{-\frac{2}{3}}$

$\begin{aligned} &\left.\frac{3 \times 3 \times 3}{2 \times 2 \times 2}\right)^{-\frac{2}{3}} \\ \Rightarrow &\left(\frac{3^{3}}{2^{3}}\right)^{-\frac{2}{3}} \\ \Rightarrow & \frac{3^{3 x}-\frac{2}{3}}{2^{2 \times-\frac{2}{3}}} \\ \Rightarrow & \frac{3^{-2}}{2^{-2}} \\ \Rightarrow & \frac{2^{2}}{3^{2}} \\ \Rightarrow & \frac{2 \times 2}{3 \times 3} \\ \Rightarrow & \frac{4}{9} \end{aligned}$

(xi)  $(0.062 .5)^{3 / 4}$

$\left(0.5 \times 0.5 \times 0.5 \times 0.5)^{\frac{3}{4}}\right.$

$\left((0.5)^{4}\right)^{\frac{-3}{4}}$

$(0.5)^{4 \times 3}$

$(0.5)^{3} \Rightarrow 0.5 \times 0.5 \times 0.5 \Rightarrow 0.125$

(xii) $\left(12 \frac{19}{27}\right)^{\frac{1}{3}}$

=$\left(\frac{12 \times 27+9}{27}\right)^{1 / 3}$

=$\left(\frac{324+19}{27}\right)^{1 / 3}$

=$\left(\frac{343}{27}\right)^{2 / 3}$

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

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

=$\frac{7^{3 \times \frac{1}{3}}}{3^{8 \times \frac{1}{3}}}$

=$\frac{7}{3}$

=$2 \frac{1}{3}$