Question

Simplify:

a.${\left(81n^{\frac{2}{5}}\right)}^{\frac{3}{2}}$b.${\left(a^{\frac{3}{2}}{b}^{\frac{1}{2}}\right)}^{\frac{4}{3}}$

Short answer

Part (a). The answer is $729n^{\frac{3}{5}}$.

Part (b). The answer is $a^2{b}^{\frac{2}{3}}$.

Step-by-step solution

Part (a) Step 1. Simplifying  

First, we use the Product to a Power Property.

So,

${\left(81\right)}^{\frac{3}{2}}.{\left(n^{\frac{2}{5}}\right)}^{\frac{3}{2}}$

To raise a power to a power, we multiply the exponents.

$$\begin{gathered} {\left(81\right)}^{\frac{3}{2}}·{\left(n^{\frac{2}{5}}\right)}^{\frac{3}{2}} \\ {3}^{4·\frac{3}{2}}·n^{\frac{2}{5}·\frac{3}{2}} \\ {3}^6·n^{\frac{3}{5}} \\ 729n^{\frac{3}{5}} \end{gathered}$$

Part (b) Step 1. Simplifying the exponent 

First, we use the Product to a Power

Property.

So,

$$\begin{gathered} {\left(a^{\frac{3}{2}}{b}^{\frac{1}{2}}\right)}^{\frac{4}{3}} \\ {a}^{\frac{3}{2}·\frac{4}{3}}{b}^{\frac{1}{2}·\frac{4}{3}} \\ {a}^2{b}^{\frac{2}{3}} \end{gathered}$$