Q 155E - Intermediate Algebra

Question

In the following exercises, simplify:

Part (a): ${\left(27q^{\frac{3}{2}}\right)}^{\frac{4}{3}}$

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

Step-by-Step Solution

Answer

Part (a): On simplifying, we get$81q^2$.

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

Part (a) Step 1. Given information.

Consider the given question,

${\left(27q^{\frac{3}{2}}\right)}^{\frac{4}{3}}$

First we use the product to a power property,

$={\left(27\right)}^{\frac{4}{3}}{\left(q^{\frac{3}{2}}\right)}^{\frac{4}{3}}$

Rewrite $27$ as a power of $3$,

$$\begin{gathered} ={\left(3^3\right)}^{\frac{4}{3}}{\left(q^{\frac{3}{2}}\right)}^{\frac{4}{3}} \\ =\left(3^{3\times \frac{4}{3}}\right)\left(q^{\frac{3}{2}\times \frac{4}{3}}\right) \\ =\left(3^4\right)\left(q^2\right) \\ =81q^2 \end{gathered}$$

Part (b) Step 1. Given information.

Consider the given question,

${\left(a^{\frac{1}{3}}{b}^{\frac{2}{3}}\right)}^{\frac{3}{2}}$

First we use the product to a power property,

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