Q. 35 - Intermediate Algebra

Question

In the following exercises, simplify using absolute values as necessary. 

$ⓐ \sqrt[3]{x^9} ⓑ \sqrt[4]{y^{12}}$

Step-by-Step Solution

Answer

Part(a) $\sqrt[3]{x^9}=x^3$

Part(b)$\sqrt[4]{y^{12}}=\left|y^3\right|$

Part(a) Step 1. Given information

$ⓐ \sqrt[3]{x^9}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \to {\left(x^3\right)}^3={\text{ x}}^9\\ \to \sqrt[3]{x^9}\\ \to \sqrt[3]{{\left(x^3\right)}^3}\\ \text{Since the index n is odd = }\sqrt[n]{a^n}=\text{ a}.\\ \sqrt[3]{x^9}=x^3\end{array}$

Part(b) Step 1. Given information

$ⓑ \sqrt[4]{y^{12}}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(y^3\right)}^4={\text{ y}}^{12}\\ \Rightarrow \sqrt[4]{y^{12}}\\ \Rightarrow \sqrt[4]{{\left(y^3\right)}^4}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ \Rightarrow \sqrt[4]{y^{12}}=\left|y^3\right|\end{array}$