Chapter 7.4, Problem 45PPS

To determine

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

Answer to Problem 45PPS

  $\frac{35-7\sqrt{3}}{22}$

Explanation of Solution

Given:

  $\frac{7}{5+\sqrt{3}}$

Formula used:

  $(a+b)(a-b)=a^2-b^2$

The radical rules:

  ${(a^m)}^n=a^{mn}$

Calculation:

  $\begin{array}{l}\Rightarrow \frac{7}{5+\sqrt{3}}\\ \Rightarrow \frac{7}{5+\sqrt{3}}\times \frac{5-\sqrt{3}}{5-\sqrt{3}}[\because \text{Rationalizing the denomiantor}]\\ \Rightarrow \frac{7(5-\sqrt{3})}{(5+\sqrt{3})(5-\sqrt{3})}\\ \Rightarrow \frac{35-7\sqrt{3}}{5^2-{(3^{\frac{1}{2}})}^2}[\because (a+b)(a-b)=a^2-b^2]\\ \Rightarrow \frac{35-7\sqrt{3}}{25-{(3)}^{\frac{2}{2}}}[\because {(}^{a^m}=a^{mn}]\end{array}$

  $\begin{array}{l}\Rightarrow \frac{35-7\sqrt{3}}{25-3}\\ \Rightarrow \frac{35-7\sqrt{3}}{22}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $\frac{35-7\sqrt{3}}{22}$ .