Chapter 7.4, Problem 50PPS

To determine

To simplify the expression $\frac{2\sqrt{5}}{2\sqrt{7}+3\sqrt{3}}$ .

Answer to Problem 50PPS

  $2\sqrt{5}(2\sqrt{7}-3\sqrt{3})$

Explanation of Solution

Given:

  $\frac{2\sqrt{5}}{2\sqrt{7}+3\sqrt{3}}$

Formula used:

  $(a+b)(a-b)=a^2-b^2$

Calculation:

  $\begin{array}{l}\Rightarrow \frac{2\sqrt{5}}{2\sqrt{7}+3\sqrt{3}}\\ \Rightarrow \frac{2\sqrt{5}}{2\sqrt{7}+3\sqrt{3}}\times \frac{2\sqrt{7}-3\sqrt{3}}{2\sqrt{7}-3\sqrt{3}}\\ \Rightarrow \frac{2\sqrt{5}(2\sqrt{7}-3\sqrt{3})}{(2\sqrt{7}+3\sqrt{3})(2\sqrt{7}-3\sqrt{3})}[\because \text{Rationalizing the denominator}]\\ \Rightarrow \frac{2\sqrt{5}(2\sqrt{7}-3\sqrt{3})}{{(2\sqrt{7})}^2-{(3\sqrt{3})}^2}[\because (a+b)(a-b)=a^2-b^2]\\ \Rightarrow \frac{2\sqrt{5}(2\sqrt{7}-3\sqrt{3})}{28-27}\\ \Rightarrow \frac{2\sqrt{5}(2\sqrt{7}-3\sqrt{3})}{1}\\ \Rightarrow 2\sqrt{5}(2\sqrt{7}-3\sqrt{3})\end{array}$

Conclusion:

The simplified form of the given expression is equal to $2\sqrt{5}(2\sqrt{7}-3\sqrt{3})$ .