Question

Simplify:${\left(\frac{64}{125}\right)}^{\frac{-2}{3}}+\frac{1}{{\left({\displaystyle \frac{256}{625}}\right)}^{{\displaystyle \frac{1}{4}}}}+{\left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)}^0$

• A. $3\frac{13}{16}$
• B. $3\frac{16}{13}$
• C. $16\frac{3}{13}$
• D. $14\frac{3}{4}$

Answer 1

The correct option is A

$3\frac{13}{16}$

Simplification

${\left(\frac{64}{125}\right)}^{\frac{-2}{3}}+\frac{1}{{\left({\displaystyle \frac{256}{625}}\right)}^{{\displaystyle \frac{1}{4}}}}+{\left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)}^0={\left({\left(\frac{4}{5}\right)}^3\right)}^{\frac{-2}{3}}+\frac{1}{{\left({\left(\frac{4}{5}\right)}^4\right)}^{\frac{1}{4}}}+1$ $\left[a^0=1\right]$

$$\begin{gathered} ={{\left(\frac{4}{5}\right)}^3}^{\times \frac{-2}{3}}+\frac{1}{{{\left(\frac{4}{5}\right)}^4}^{\times \frac{1}{4}}}+1\left[{\left(a^x\right)}^y=a^{x\times y}\right] \\ ={\left(\frac{4}{5}\right)}^{-2}+\left(\frac{5}{4}\right)+1 \\ ={\left(\frac{5}{4}\right)}^2+\left(\frac{5}{4}\right)+1 \end{gathered}$$

$$\begin{gathered} =\frac{25}{16}+\frac{5}{4}+1 \\ =\frac{25+20+16}{16}=\frac{61}{16}=3\frac{13}{16} \end{gathered}$$

Therefore, ${\left(\frac{64}{125}\right)}^{\frac{-2}{3}}+\frac{1}{{\left({\displaystyle \frac{256}{625}}\right)}^{{\displaystyle \frac{1}{4}}}}+{\left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)}^0$ is equal to $3\frac{13}{16}$

Hence, option (A) is correct.