Chapter 7.4, Problem 57PPS

To determine

To simplify the expression

  $(\sqrt{3}-\sqrt{2})(\sqrt{15}+\sqrt{12})$ .

Answer to Problem 57PPS

  $(3-\sqrt{6})(\sqrt{5}+2)$

Explanation of Solution

Given:

  $(\sqrt{3}-\sqrt{2})(\sqrt{15}+\sqrt{12})$

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{3}-\sqrt{2})(\sqrt{15}+\sqrt{12})\\ \Rightarrow (\sqrt{3}-\sqrt{2})(\sqrt{5\times 3}+\sqrt{2\times 2\times 3})\\ \Rightarrow (\sqrt{3}-\sqrt{2})(\sqrt{5}\times \sqrt{3}+\sqrt{2\times 2}\times \sqrt{3})[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\\ \Rightarrow (\sqrt{3}-\sqrt{2})(\sqrt{5}\times \sqrt{3}+\sqrt{2^2}\times \sqrt{3})\\ $ \Rightarrow (\sqrt{3}-\sqrt{2})(\sqrt{5}\times \sqrt{3}+{(2^2)}^{\frac{1}{2}}\times \sqrt{3})[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow (\sqrt{3}-\sqrt{2})(\sqrt{5}\times \sqrt{3}+{(2)}^{\frac{2}{2}}\times \sqrt{3})[\because {(a^m)}^n=a^{mn}]$\Rightarrow (\sqrt{3}-\sqrt{2})(\sqrt{5}\times \sqrt{3}+(2)\times \sqrt{3})\\ \Rightarrow (\sqrt{3}-\sqrt{2})\times \sqrt{3}\times (\sqrt{5}+2)\\ \Rightarrow (\sqrt{3}\times \sqrt{3}-\sqrt{3}\times \sqrt{2})(\sqrt{5}+2)\\ \Rightarrow (3-\sqrt{6})(\sqrt{5}+2)\end{array}$

Conclusion:

The simplified form of the given expression is equal to $(3-\sqrt{6})(\sqrt{5}+2)$ .