Q. 8.67 - Intermediate Algebra

Question

Simplify:

a. ${\left(32x^{\frac{1}{3}}\right)}^{\frac{3}{5}}$ b. ${\left(x^{\frac{3}{4}}{y}^{\frac{1}{2}}\right)}^{\frac{2}{3}}$

Step-by-Step Solution

Answer

Part (a). The answer is $8·x^{\frac{1}{5}}$.

part (b). The answer is  $\left(x^{\frac{1}{2}}{y}^{\frac{1}{3}}\right)$.

Part (a) Step 1. Simplifying

First, we use the Product to a Power

Property.

So,

$$\begin{gathered} {\left(32x^{\frac{1}{3}}\right)}^{\frac{3}{5}} \\ {\left(32\right)}^{\frac{3}{5}}·{\left(x^{\frac{1}{3}}\right)}^{\frac{3}{5}} \end{gathered}$$

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

$$\begin{gathered} {\left(32\right)}^{\frac{3}{5}}·{\left(x^{\frac{1}{3}}\right)}^{\frac{3}{5}} \\ {\left(2^5\right)}^{\frac{3}{5}}·{\left(x^{\frac{1}{3}}\right)}^{\frac{3}{5}} [32=2^5] \\ {2}^3·x^{\frac{1}{5}} \\ 8·x^{\frac{1}{5}} [2^3=8] \end{gathered}$$

Part (b) Step 1. Simplifying the exponent

First, we use the Product to a

Power Property

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

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

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