Q. 8.17 - Intermediate Algebra

Question

Simplify: $\left(a\right) \sqrt[4]{u^{12}} \left(b\right) \sqrt[3]{v^{15}}$

Step-by-Step Solution

Answer

The solution of (a) and (b) are $\left|u^3\right| and v^5$ respectively.

Part (a) Step 1. Given information

The root: $\sqrt[4]{u^{12}}$

Part (a) Step 2. Calculation

We have,

$$\begin{gathered} \sqrt[4]{u^{12}} \\ =\sqrt[4]{{\left(u^3\right)}^4} \sin ce {\left(u^3\right)}^4=u^{12} \\ =\left|u^3\right| \sin ce the index n is even \sqrt[n]{a^n}=\left|a\right| \end{gathered}$$

Therefore, the solution is $\left|u^3\right|$

Part (b) Step 1. Given information

The root: $\sqrt[3]{v^{15}}$

Part (b) Step 2. Calculation

We have,

$$\begin{gathered} \sqrt[3]{v^{15}} \\ =\sqrt[3]{{\left(v^5\right)}^3} \sin ce {\left(v^5\right)}^3=v^{15} \\ =v^5 \sin ce n is odd, \sqrt[n]{a^n}=a \end{gathered}$$

Therefore, the solution is $v^5$