Question

Find the value of $(\frac{729}{512}{)}^{-\frac{2}{3}}$.

• A. $\frac{81}{64}$
  
• B. $\frac{64}{81}$
  
• C. $\frac{9}{8}$
  
• D. $\frac{8}{9}$
  

Answer 1

The correct option is B $\frac{64}{81}$


$\underset{–}{\text{Need to Find:}}$ $(\frac{729}{512}{)}^{-\frac{2}{3}}$

To eliminate the negative sign from the exponent, we would take the fraction's (base) reciprocal.

$(\frac{729}{512}{)}^{-\frac{2}{3}}=(\frac{512}{729}{)}^{\frac{2}{3}}$
$[\because (\frac{a}{b}{)}^{-n}=(\frac{b}{a}{)}^n]$

Now,
$(\frac{512}{729}{)}^{\frac{2}{3}}$

$=(\frac{8^3}{9^3}{)}^{\frac{2}{3}}$
$(\because 512=8\times 8\times 8=8^3\text{and}729=9\times 9\times 9=9^3)$

$=[(\frac{8}{9}{)}^3{]}^{\frac{2}{3}}$
$(\because \frac{a^n}{b^n}=(\frac{a}{b}{)}^n)$


$=(\frac{8}{9}{)}^{3\times \frac{2}{3}}$
$(\because (a^m{)}^n=a^{m\times n})$

$=(\frac{8}{9}{)}^2$

$=\frac{8^2}{9^2}$
$(\because (\frac{a}{b}{)}^n=\frac{a^n}{b^n})$

$=\frac{8\times 8}{9\times 9}$

$\Rightarrow \underset{–}{(\frac{512}{729}{)}^{\frac{2}{3}}=(\frac{729}{512}{)}^{-\frac{2}{3}}=\frac{64}{81}}$


Hence, the value of $(\frac{729}{512}{)}^{-\frac{2}{3}}$ is $\frac{64}{81}$.

$\therefore$ Option (b.) is the correct answer.