Q 157E - Intermediate Algebra

Question

In the following exercises, simplify:

Part (a): ${\left(16u^{\frac{1}{3}}\right)}^{\frac{3}{4}}$

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

Step-by-Step Solution

Answer

Part (a): On simplifying, we get $8u^{\frac{1}{4}}$.

Part (b): On simplifying, we get $8p^{\frac{1}{2}}{q}^{\frac{3}{4}}$.

Part (a) Step 1. Given information.

Consider the given question,

${\left(16u^{\frac{1}{3}}\right)}^{\frac{3}{4}}$

First we use the product to a power property,

$={\left(16\right)}^{\frac{3}{4}}{\left(u^{\frac{1}{3}}\right)}^{\frac{3}{4}}$

Rewrite $16$ as a power of $2$,

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

Part (b) Step 1. Given information.

Consider the given question,

${\left(4p^{\frac{1}{3}}{q}^{\frac{1}{2}}\right)}^{\frac{3}{2}}$

First we use the product to a power property,

$={\left(4\right)}^{\frac{3}{2}}{\left(p^{\frac{1}{3}}\right)}^{\frac{3}{2}}{\left(q^{\frac{1}{2}}\right)}^{\frac{3}{2}}$

Rewrite $4$ as a power of $2$,

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