Q. 77 - Intermediate Algebra

Question

In the following exercises, simplify using absolute value signs as needed. $\sqrt{192q^3{r}^7},\sqrt[3]{54m^9{n}^{10}},\sqrt[4]{81a^9{b}^8}$

Step-by-Step Solution

Answer

The expression in simplified form is $8\left|q{r}^3\right|\sqrt{3qr},3m^3{n}^3\sqrt[3]{2n},3a^2{b}^2\sqrt[4]{a}$ respectively

Step 1: Given information

Given the expression $\sqrt{192q^3{r}^7},\sqrt[3]{54m^9{n}^{10}},\sqrt[4]{81a^9{b}^8}$

Step 2: Using the product property

Using the property, we get

$$\begin{gathered} \sqrt{192q^3{r}^7}=\sqrt{64q^2{r}^6·3qr} \\ \sqrt{192q^3{r}^7}=\sqrt{{\left(8q{r}^3\right)}^2·3qr} \\ \sqrt{192q^3{r}^7}=\sqrt{{\left(8q{r}^3\right)}^2}·\sqrt{3qr} \\ \sqrt{192q^3{r}^7}=8\left|q{r}^3\right|\sqrt{3qr} \end{gathered}$$

Step 3: Using the product property

Using the property, we get

$$\begin{gathered} \sqrt[3]{54m^9{n}^{10}}=\sqrt[3]{27m^9{n}^9·2n} \\ \sqrt[3]{54m^9{n}^{10}}=\sqrt[3]{{\left(3m^3{n}^3\right)}^3·2n} \\ \sqrt[3]{54m^9{n}^{10}}=\sqrt[3]{{\left(3m^3{n}^3\right)}^3}·\sqrt[3]{2n} \\ \sqrt[3]{54m^9{n}^{10}}=3m^3{n}^{3 . }\sqrt[ 3]{2n} \end{gathered}$$

Step 4: Using the product property

Using the property, we get

$$\begin{gathered} \sqrt[4]{81a^9{b}^8}=\sqrt[4]{81a^8{b}^8·a} \\ \sqrt[4]{81a^9{b}^8}=\sqrt[4]{{\left(3a^2{b}^2\right)}^4·a} \\ \sqrt[4]{81a^9{b}^8}==\sqrt[4]{{\left(3a^2{b}^2\right)}^4}·\sqrt[4]{a} \\ \sqrt[4]{81a^9{b}^8}=3\left|a^2{b}^2\right|.\sqrt[4]{a} \\ \sqrt[4]{81a^9{b}^8}=3a^2{b}^2.\sqrt[4]{a} \end{gathered}$$