Chapter 7.4, Problem 69PPS

To determine

To simplify the expression $2\sqrt{\frac{1}{2}}+2\sqrt{2}-\sqrt{8}$ .

Answer to Problem 69PPS

  $\sqrt{2}$

Explanation of Solution

Given:

  $2\sqrt{\frac{1}{2}}+2\sqrt{2}-\sqrt{8}$

Formula used:

The radical rules:

  $\sqrt[n]{a}=a^{\frac{1}{n}}$

  ${(a^m)}^n=a^{mn}$

  $\sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}$

  $a^m\cdot a^n=a^{m+n}$

Calculation:

  $\begin{array}{l}\Rightarrow 2\sqrt{\frac{1}{2}}+2\sqrt{2}-\sqrt{8}\\ \Rightarrow 2\frac{\sqrt{1}}{\sqrt{2}}+2\sqrt{2}-\sqrt{8}[\because \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}]\\ \Rightarrow \frac{2}{\sqrt{2}}+2\sqrt{2}-\sqrt{8}\text{ [}\because \sqrt{1}=1]\\ \Rightarrow \frac{\sqrt{2}\times \sqrt{2}}{\sqrt{2}}+2\sqrt{2}-\sqrt{8}\text{ [}\because \text{2 =}\sqrt{2}\times \sqrt{2}\text{]}\\ \Rightarrow \sqrt{2}+2\sqrt{2}-\sqrt{8}\\ \Rightarrow \sqrt{2}+2\sqrt{2}-\sqrt{2\times 2\times 2}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\\ \Rightarrow \sqrt{2}+2\sqrt{2}-\sqrt{2^2}\times \sqrt{2}\\ $ \Rightarrow \sqrt{2}+2\sqrt{2}-{(2^2)}^{\frac{1}{2}}\times \sqrt{2}[\because {(a^m)}^n=a^{mn}]$\Rightarrow \sqrt{2}+2\sqrt{2}-2^1\times \sqrt{2}\\ \Rightarrow \sqrt{2}+2\sqrt{2}-2\sqrt{2}\\ \Rightarrow \sqrt{2}\end{array}$

Conclusion:

The simplified form of the given expression is equal to

  $\sqrt{2}.$