Chapter 7.4, Problem 54PPS

To determine

To simplify the expression $3\sqrt{50}-3\sqrt{32}$ .

Answer to Problem 54PPS

  $3\sqrt{2}$

Explanation of Solution

Given:

  $3\sqrt{50}-3\sqrt{32}$

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 3\sqrt{50}-3\sqrt{32}\\ \Rightarrow 3\times \sqrt{5\times 5\times 2}-3\times \sqrt{2\times 2\times 2\times 2\times 2}\\ \Rightarrow 3\times \sqrt{2}\times \sqrt{5\times 5}-3\times \sqrt{2\times 2\times 2\times 2\times 2}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\\ \Rightarrow 3\times \sqrt{2}\times \sqrt{{(5)}^2}-3\times \sqrt{{(2)}^5}\\ $ \Rightarrow 3\times {(2)}^{\frac{1}{2}}\times {(5^2)}^{\frac{1}{2}}-3\times {(2^5)}^{\frac{1}{2}}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow 3\times {(2)}^{\frac{1}{2}}\times {(5)}^{\frac{2}{2}}-3\times {(2)}^{\frac{5}{2}}[\because {(a^m)}^n=a^{mn}]$\Rightarrow 3\times \sqrt{2}\times 5-3\times 4\times \sqrt{2}\\ \Rightarrow 3\sqrt{2}(5-4)\\ \Rightarrow 3\sqrt{2}\end{array}$

Conclusion:

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