Q. 194 - Intermediate Algebra

Question

In the following exercises, multiply.

a.$\sqrt{2}(-5+9\sqrt{2})$

b.$\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})$

Step-by-Step Solution

Answer

On multiplication, we get $-5\sqrt{2}+18,\sqrt[4]{24}+2\sqrt[4]{3}$ respectively

Part (a) Step 1: Given information

Given the expression $\sqrt{2}(-5+9\sqrt{2}),\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})$

Part (a) Step 2: Multiplying the terms and simplifying

Multiplying and simplifying we get

$$\begin{gathered} \sqrt{2}(-5+9\sqrt{2})=\sqrt{2}·(-5)+9\sqrt{(2{)}^2} \\ \sqrt{2}(-5+9\sqrt{2})=\sqrt{2}·(-5)+9·2 \\ \sqrt{2}(-5+9\sqrt{2})=-5\sqrt{2}+18 \end{gathered}$$

Part (b) Step 1: Given information

Given expression $\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})$

Part (b) Step 2: Multiplying the terms and simplifying

Multiplying the terms and simplifying, we get

$$\begin{gathered} \sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})=\sqrt[4]{2}·\sqrt[4]{12}+\sqrt[4]{2}·\sqrt[4]{24} \\ \sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})=\sqrt[4]{24}+\sqrt[4]{48} \\ \sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})=\sqrt[4]{24}+\sqrt[4]{16·3} \\ \sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})=\sqrt[4]{24}+\sqrt[4]{16}·\sqrt[4]{3} \\ \sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})=\sqrt[4]{24}+\sqrt[4]{(2{)}^4}·\sqrt[4]{3} \\ \sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})=\sqrt[4]{24}+2\sqrt[4]{3} \end{gathered}$$