Q 156E - Intermediate Algebra

Question

In the following exercises, simplify:

Part (a): ${\left(64s^{\frac{3}{7}}\right)}^{\frac{1}{6}}$

Part (b):  ${\left(m^{\frac{4}{3}}{n}^{\frac{1}{2}}\right)}^{\frac{3}{4}}$

Step-by-Step Solution

Answer

Part (a): On simplifying, we get $2s^{\frac{1}{14}}$.

Part (b): On simplifying, we get $m{n}^{\frac{3}{8}}$.

Part (a) Step 1. Given information.

Consider the given question,

${\left(64s^{\frac{3}{7}}\right)}^{\frac{1}{6}}$

First we use the product to a power property,

$={\left(64\right)}^{\frac{1}{6}}{\left(s^{\frac{3}{7}}\right)}^{\frac{1}{6}}$

Rewrite $64$ as a power,

$$\begin{gathered} ={\left(2^6\right)}^{\frac{1}{6}}{\left(s^{\frac{3}{7}}\right)}^{\frac{1}{6}} \\ =\left(2^{6\times \frac{1}{6}}\right)\left(s^{\frac{3}{7}\times \frac{1}{6}}\right) \\ =2s^{\frac{1}{14}} \end{gathered}$$

Part (b) Step 1. Given information.

Consider the given question,

${\left(m^{\frac{4}{3}}{n}^{\frac{1}{2}}\right)}^{\frac{3}{4}}$

First we use the product to a power property,

$$\begin{gathered} ={\left(m^{\frac{4}{3}}\right)}^{\frac{3}{4}}{\left(n^{\frac{1}{2}}\right)}^{\frac{3}{4}} \\ =\left(m^{\frac{4}{3}\times \frac{3}{4}}\right)\left(n^{\frac{1}{2}\times \frac{3}{4}}\right) \\ =m{n}^{\frac{3}{8}} \end{gathered}$$