Chapter 7.4, Problem 65PPS

To determine

To simplify the expression $\sqrt{2}(\sqrt{8}+\sqrt{6})$ .

Answer to Problem 65PPS

  $4+2\sqrt{3}$

Explanation of Solution

Given:

  $\sqrt{2}(\sqrt{8}+\sqrt{6})$

Formula used:

The radical rules:

  $\sqrt[n]{a}=a^{\frac{1}{n}}$

  ${(a^m)}^n=a^{mn}$

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

Calculation:

  $\begin{array}{l}\Rightarrow \sqrt{2}(\sqrt{8}+\sqrt{6})\\ \Rightarrow \sqrt{2}(\sqrt{2\times 2\times 2}+\sqrt{2\times 3})\\ \Rightarrow \sqrt{2\times 2\times 2\times 2}+\sqrt{2\times 2}\times \sqrt{3}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\\ \Rightarrow \sqrt{{(2)}^4}+\sqrt{{(2)}^2}\times \sqrt{3}\\ $ \Rightarrow {(2^4)}^{\frac{1}{2}}+{(2^2)}^{\frac{1}{2}}\times \sqrt{3}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow 2^{\frac{4}{2}}+{(2)}^{\frac{2}{2}}\times \sqrt{3}[\because {(a^m)}^n=a^{mn}]$\Rightarrow 2^2+2\times \sqrt{3}\\ \Rightarrow 4+2\sqrt{3}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $4+2\sqrt{3}$ .