Q. 74 - Intermediate Algebra

Question

In the following exercises, simplify using absolute value signs as needed. $\sqrt{175n^{13}},\sqrt[5]{512p^5},\sqrt[4]{324q^7}$

Step-by-Step Solution

Answer

The expression on simplification is $5n^6\sqrt{7n},2p\sqrt[5]{16},\left|3q\right|\sqrt[4]{4q^3}$ respectively

Step 1: Given information

Given the expression $\sqrt{175n^{13}},\sqrt[5]{512p^5},\sqrt[4]{324q^7}$

Step 2: Using the product property

Using the property, we get

$$\begin{gathered} \sqrt{175n^{13}}=\sqrt{25n^{12}·7n} \\ \sqrt{175n^{13}}=\sqrt{{\left(5n^6\right)}^2·7n} \\ \sqrt{175n^{13}}=\sqrt{{\left(5n^6\right)}^2}·\sqrt{7n} \\ \sqrt{175n^{13}}=5\left|n^6\right|\sqrt{7n} \\ \sqrt{175n^{13}}=5n^6\sqrt{7n} \end{gathered}$$

Step 3: Using the product property

Using the property, we get

$$\begin{gathered} \sqrt[5]{512p^5}=\sqrt[5]{32p^5·16} \\ \sqrt[5]{512p^5}=\sqrt[5]{(2p{)}^5·16} \\ \sqrt[5]{512p^5}=\sqrt[5]{(2p{)}^5}·\sqrt[5]{16} \\ \sqrt[5]{512p^5}=2p\sqrt[5]{16} \end{gathered}$$

Step 4: Using the product property

Using the property, we have

$$\begin{gathered} \sqrt[4]{324q^7}=\sqrt[4]{81q^4·4q^3} \\ \sqrt[4]{324q^7}=\sqrt[4]{(3q{)}^4·4q^3} \\ \sqrt[4]{324q^7}=\sqrt[4]{(3q{)}^4}·\sqrt[4]{4q^3} \\ \sqrt[4]{324q^7}=\left|3q\right|\sqrt[4]{4q^3} \end{gathered}$$