Chapter CSB, Problem 2.39E

To determine

To simplify: the given square root.

Answer to Problem 2.39E

  $\frac{4}{4-3\sqrt{3}}=\frac{-4}{11}\left(4+3\sqrt{3}\right)$ .

Explanation of Solution

Given:

The given square rootis $\frac{4}{4-3\sqrt{3}}$ .

Calculation:

As the given square root is $\frac{4}{4-3\sqrt{3}}$ .

Now, to simplify the square root, therefore,

  $\begin{array}{l}\frac{4}{4-3\sqrt{3}}=\frac{4}{4-3\sqrt{3}}\cdot \left(\frac{4+3\sqrt{3}}{4+3\sqrt{3}}\right)\left\{\text{conjugate multiplication}\right\}\\ \frac{4}{4-3\sqrt{3}}=\frac{4\left(4+3\sqrt{3}\right)}{\left(4+3\sqrt{3}\right)\left(4-3\sqrt{3}\right)}\left\{\text{simplify}\right\}\\ \{\text{expand as}\left(a+b\right)\left(a-b\right)=a^2-b^2\}\\ \therefore \frac{4}{4-3\sqrt{3}}=\frac{4\left(4+3\sqrt{3}\right)}{{\left(4\right)}^2-{\left(3\sqrt{3}\right)}^2}\\ \Rightarrow \frac{4}{4-3\sqrt{3}}=\frac{4\left(4+3\sqrt{3}\right)}{16-27}\\ \Rightarrow \frac{4}{4-3\sqrt{3}}=\frac{4}{-11}\left(4+3\sqrt{3}\right)\left\{\text{multiply by }\right\}\\ \therefore \frac{4}{4-3\sqrt{3}}=\frac{-4}{11}\left(4+3\sqrt{3}\right)\end{array}$

Hence, the simplification of the given function is $\frac{4}{4-3\sqrt{3}}=\frac{-4}{11}\left(4+3\sqrt{3}\right)$ .