Q. 173 - Intermediate Algebra

Question

In the following exercises, simplify. 

a.$\sqrt{27}-\sqrt{75}$

b.$\sqrt[3]{40}-\sqrt[3]{320}$

c.$\frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}$

Step-by-Step Solution

Answer

The simplified form is $-2\sqrt{3},-2\sqrt[3]{5},3\sqrt[4]{2}$ respectively

Part (a) Step 1: Given information

Given the expression

 $\sqrt{27}-\sqrt{75}$$\sqrt{27}-\sqrt{75},\sqrt[3]{40}-\sqrt[3]{320},\frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}$

Part (a) Step 2: Simplifying the radical and combining the like term

Simplifying and combining the like terms, we get

$$\begin{gathered} \sqrt{27}-\sqrt{75}=\sqrt{9\times 3}-\sqrt{25\times 3} \\ \sqrt{27}-\sqrt{75}=3\sqrt{3}-5\sqrt{3} \\ \sqrt{27}-\sqrt{75}=-2\sqrt{3} \end{gathered}$$

Part (b) Step 1: Given expression

Given expression $\sqrt[3]{40}-\sqrt[3]{320}$

Part (b) Step 2: Simplifying and combining the like terms,

Simplifying and combining the like terms, we get

$$\begin{gathered} \sqrt[3]{40}-\sqrt[3]{320}=\sqrt[3]{8·5}-\sqrt[3]{64·5} \\ =\sqrt[3]{8}·\sqrt[3]{5}-\sqrt[3]{64}·\sqrt[3]{5} \\ =2\sqrt[3]{5}-4\sqrt[3]{5} \\ =-2\sqrt[3]{5} \end{gathered}$$

Part (c) Step 1: Given information

Given expression $\frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}$

Part (c) Step 2: Simplifying and combining the like terms, we get

Simplifying, we get

$$\begin{gathered} \frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}=\frac{1}{2}·2\sqrt[4]{2}+\frac{2}{3}\sqrt[4]{2}·3 \\ \frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}=\frac{1}{2}·2\sqrt[4]{2}+\frac{2}{3}\sqrt[4]{2}·3 \\ \frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}=\sqrt[4]{2}+2\sqrt[4]{2} \\ \frac{1}{2}\sqrt[4]{32}+\frac{2}{3}\sqrt[4]{162}=3\sqrt[4]{2} \end{gathered}$$