Q. 70 - Intermediate Algebra

Question

In the following exercises, simplify using absolute value signs as needed. $\sqrt{r^{25}},\sqrt[5]{p^8},\sqrt[4]{m^5}$

Step-by-Step Solution

Answer

The expression in simplified form is $r^{12}\sqrt{r},p\sqrt[5]{p^3},\left|m\right|\sqrt[4]{m}$ respectively

Step 1: Given information

Given the expression $\sqrt{r^{25}},\sqrt[5]{p^8},\sqrt[4]{m^5}$

Step 2: Using the product property

Using the product property, we get

$$\begin{gathered} \sqrt{r^{25}}=\sqrt{{\left(r^{12}\right)}^2·r^1} \\ \sqrt{r^{25}}=\sqrt{{\left(r^{12}\right)}^2·r} \\ \sqrt{r^{25}}=\sqrt{{\left(r^{12}\right)}^2}·\sqrt{r} \\ \sqrt{r^{25}}=\left|r^{12}\right|\sqrt{r} \\ \sqrt{r^{25}}=r^{12}\sqrt{r} \end{gathered}$$

Step 3; Using the product property

Using the property, we get

$$\begin{gathered} \sqrt[5]{p^8}=\sqrt[5]{p^5·p^3} \\ \sqrt[5]{p^8}==\sqrt[5]{p^5}·\sqrt[5]{p^3} \\ \sqrt[5]{p^8}=p\sqrt[5]{p^3} \end{gathered}$$

Step 4: Using the product property

Using the property, we get

$$\begin{gathered} \sqrt[4]{m^5}=\sqrt[4]{{\left(m^1\right)}^4·m^1} \\ \sqrt[4]{m^5}=\sqrt[4]{{\left(m^1\right)}^4·m} \\ \sqrt[4]{m^5}=\sqrt[4]{{\left(m^1\right)}^4}·\sqrt[4]{m} \\ \sqrt[4]{m^5}=\left|m\right|\sqrt[4]{m} \end{gathered}$$