Q 176E - Intermediate Algebra

Question

In the following exercises, simplify.

a.$\sqrt{45}+\sqrt{80}$

b.$\sqrt[3]{81}-\sqrt[3]{192}$

c.$\frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}$

Step-by-Step Solution

Answer

On simplification, we get $7\sqrt{5},-\sqrt[3]{3},12\sqrt[4]{5}$ respectively

Step 1: Given information

Given the expression $\sqrt{45}+\sqrt{80},\sqrt[3]{81}-\sqrt[3]{192},\frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}$

Step 2: Simplifying the radical and combine like terms

Simplifying radical and combining we get

$$\begin{gathered} \sqrt{45}+\sqrt{80}=\sqrt{9·5}+\sqrt{16·5} \\ \sqrt{45}+\sqrt{80}=3\sqrt{5}+4\sqrt{5} \\ \sqrt{45}+\sqrt{80}==7\sqrt{5} \end{gathered}$$

Step 3: Simplifying the radical and combine like terms

Simplifying radical and combining we get

$$\begin{gathered} \sqrt[3]{81}-\sqrt[3]{192}=\sqrt[3]{27·3}-\sqrt[3]{64·3} \\ \sqrt[3]{81}-\sqrt[3]{192}=\sqrt[3]{27}·\sqrt[3]{3}-\sqrt[3]{64}·\sqrt[3]{3} \\ \sqrt[3]{81}-\sqrt[3]{192}=3\sqrt[3]{3}-4\sqrt[3]{3} \\ \sqrt[3]{81}-\sqrt[3]{192}=-\sqrt[3]{3} \end{gathered}$$

Step 4: Simplifying the radical and combine like terms

Simplifying radical and combining we get

$$\begin{gathered} \frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}=\frac{5}{2}·\sqrt[4]{16·5}+\frac{7}{3}·\sqrt[4]{81·5} \\ \frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}=\frac{5}{2}·\sqrt[4]{16}·\sqrt[4]{5}+\frac{7}{3}·\sqrt[4]{81}·\sqrt[4]{5} \\ \frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}=\frac{5}{2}·2·\sqrt[4]{5}+\frac{7}{3}·3·\sqrt[4]{5} \\ \frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}=\frac{10}{2}·\sqrt[4]{5}+\frac{21}{3}·\sqrt[4]{5} \\ \frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}=5\sqrt[4]{5}+7\sqrt[4]{5} \\ \frac{5}{2}\sqrt[4]{80}+\frac{7}{3}\sqrt[4]{405}=12\sqrt[4]{5} \end{gathered}$$