Chapter CSB, Problem 2.34E

To determine

To simplify: the given square root.

Answer to Problem 2.34E

  $\frac{7}{3-\sqrt{7}}=\frac{7}{2}\left(3+\sqrt{7}\right)$ .

Explanation of Solution

Given:

The given square rootis $\frac{7}{3-\sqrt{7}}$ .

Calculation:

As the given square root is $\frac{7}{3-\sqrt{7}}$ .

Now, to simplify the square root, therefore,

  $\begin{array}{l}\frac{7}{3-\sqrt{7}}=\frac{7}{3-\sqrt{7}}\cdot \left(\frac{3+\sqrt{7}}{3+\sqrt{7}}\right)\left\{\text{conjugate multiplication}\right\}\\ \frac{7}{3-\sqrt{7}}=\frac{7\left(3+\sqrt{7}\right)}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}\left\{\text{simplify}\right\}\\ \{\text{expand as}\left(a+b\right)\left(a-b\right)=a^2-b^2\}\\ \therefore \frac{7}{3-\sqrt{7}}=\frac{7\left(3+\sqrt{7}\right)}{{\left(3\right)}^2-{\left(\sqrt{7}\right)}^2}\\ \Rightarrow \frac{7}{3-\sqrt{7}}=\frac{7\left(3+\sqrt{7}\right)}{9-7}\\ \Rightarrow \frac{7}{3-\sqrt{7}}=\frac{7}{2}\left(3+\sqrt{7}\right)\end{array}$

Hence, the simplification of the given function is $\frac{7}{3-\sqrt{7}}=\frac{7}{2}\left(3+\sqrt{7}\right)$ .