Q. 64 - Intermediate Algebra

Question

In the following exercises, use the Product Property to simplify radical expressions a. $\sqrt[3]{625}$, b. $\sqrt[6]{128}$

Step-by-Step Solution

Answer

The expression in simplified form is $5\sqrt[3]{5},2\sqrt[6]{2}$ respectively

Step 1: Given information

Given the expression $\sqrt[3]{625},\sqrt[6]{128}$

Step 2: Using the product property to simplify

Using the property, we get

$$\begin{gathered} \sqrt[3]{625}=\sqrt[3]{125·5} \\ \sqrt[3]{625}=\sqrt[5]{125}\times \sqrt[3]{5} \\ \sqrt[3]{625}=\sqrt[3]{5^3}\times \sqrt[3]{5} \\ \sqrt[3]{625}=5\sqrt[3]{5} \end{gathered}$$

Step 3: Checking for another expression

Using the property, we get

$$\begin{gathered} \sqrt[6]{128}=\sqrt[6]{64\times 2} \\ \sqrt[6]{128}=\sqrt[6]{64}\times \sqrt[6]{2} \\ \sqrt[6]{128}=\sqrt[6]{2^6}\times \sqrt[6]{2} \\ \sqrt[6]{128}=2\sqrt[6]{2} \end{gathered}$$