Q. 8.15 - Intermediate Algebra

Question

Simplify: $\left(a\right) \sqrt{y^{18}} \left(b\right) \sqrt{z^{12}}$

Step-by-Step Solution

Answer

The solutions of (a) and (b) are $\left|y^9\right| and z^6$ respectively.

Part (a) Step 1. Given information

The root: $\sqrt{y^{18}}$

Part (a) Step 2. Calculation

We have,

$$\begin{gathered} \sqrt{y^{18}} \\ =\sqrt{{\left(y^9\right)}^2} \sin ce {\left(y^9\right)}^2=y^{18} \\ =\left|y^9\right| \sin ce the index n is even \sqrt{a^n}=\left|a\right| \end{gathered}$$

Therefore, the solution is $\left|y^9\right|$

Part (b) Step 2. Given information

The root: $\sqrt{z^{12}}$

Part (b) Step 2. Calculation

We have,

$$\begin{gathered} \sqrt{z^{12}} \\ =\sqrt{{\left(z^6\right)}^2} \sin ce {\left(z^6\right)}^2=z^{12} \\ =z^6 \end{gathered}$$

In this case, the absolute value sign is not needed as $z^6$ is positive.

Therefore, the solution is $z^6$