Q. 8.14 - Intermediate Algebra

Question

Simplify: $\left(a\right) \sqrt{y^2} \left(b\right) \sqrt[3]{p^3} \left(c\right) \sqrt[4]{z^4} \left(d\right) \sqrt[5]{q^5}$

Step-by-Step Solution

Answer

The solutions of (a), (b), (c) and (d) are $\left|y\right|, p, \left|z\right| and q$ respectively.

Part (a) Step 1. Given information

The root: $\sqrt{y^2}$

Part (a) Step 2. Calculation

We use the absolute value to be sure to get the positive root.

$$\begin{gathered} \sqrt{y^2} \\ =\left|y\right| \end{gathered}$$

Since the index n is even, $\sqrt[2]{a^n}=\left|a\right|$

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

Part (b) Step 1. Given information

The root: $\sqrt[3]{p^3}$

Part (b) Step 2. Calculation

This is an odd indexed root so there is no need for an absolute value sign.

$$\begin{gathered} \sqrt[3]{p^3} \\ =p \end{gathered}$$

Since the index n is odd, $\sqrt[n]{a^n}=a$

Therefore, the solution is $p$

Part (c) Step 1. Given information

The root: $\sqrt[4]{z^4}$

Part (c) Step 2. Calculation

We use the absolute value to be sure to get the positive root.

$$\begin{gathered} \sqrt[4]{z^4} \\ =\left|z\right| \end{gathered}$$

Since the index n is even, $\sqrt[n]{a^n}=\left|a\right|$

The solution is $\left|z\right|$

Part (d) Step 1. Given information

The root: $\sqrt[5]{q^5}$

Part (d) Step 2. Calculation

This is an odd indexed root so there is no need for an absolute value sign.

$$\begin{gathered} \sqrt[5]{q^5} \\ =q \end{gathered}$$

Since the index n is odd, $\sqrt[n]{a^n}=a$

Therefore, the solution is $q$