Question

Simplify: $(0.00032{)}^{3/5}\times {\left(\frac{8}{125}\right)}^{-\frac{4}{3}}$.

Answer 1

Step 1: Write in exponent form

We have $(0.00032{)}^{3/5}\times {\left(\frac{8}{125}\right)}^{-\frac{4}{3}}$

It can be written as :

${\left(\frac{32}{100000}\right)}^{\frac{3}{5}}\times {\left(\frac{8}{125}\right)}^{-\frac{4}{3}}$

Also we know that $32=2^5,8=2^3\&125=5^3$ .

Step 2: Substituting values

$$\begin{gathered} {\left(\frac{32}{100000}\right)}^{\frac{3}{5}}\times {\left(\frac{8}{125}\right)}^{-\frac{4}{3}}={\left(\frac{2^5}{{10}^5}\right)}^{\frac{3}{5}}\times {\left(\frac{2^3}{5^3}\right)}^{-\frac{4}{3}} \\ ={\left(\frac{2}{10}\right)}^{5\times \frac{3}{5}}\times {\left(\frac{2}{5}\right)}^{3\times -\frac{4}{3}} \\ ={\left(\frac{2}{10}\right)}^3\times {\left(\frac{2}{5}\right)}^{-4} \\ ={\left(\frac{2}{10}\right)}^3\times {\left(\frac{5}{2}\right)}^4 \\ ={\left(\frac{1}{5}\right)}^3\times {\left(\frac{5}{2}\right)}^4 \\ =\left(\frac{5}{2^4}\right) \\ =\frac{5}{16} \end{gathered}$$

So, $(0.00032{)}^{3/5}\times {\left(\frac{8}{125}\right)}^{-\frac{4}{3}}$$=\frac{5}{16}$