Chapter 0, Problem 63PFG

To determine

To simplify the expression

Answer to Problem 63PFG

  $\frac{12+3\sqrt{5}}{11}$ is the simplified equation.

Explanation of Solution

Given information:

  $\frac{3}{4-\sqrt{5}}$

Calculation:

The given expression is $\frac{3}{4-\sqrt{5}}$ . The denominator is irrational making the expression irrational.

To solve the expression, multiply the equation by $\frac{4+\sqrt{5}}{4+\sqrt{5}}$

= $\frac{3}{4-\sqrt{5}}\times \frac{4+\sqrt{5}}{4+\sqrt{5}}$

So,

= $\frac{3(4+\sqrt{5})}{(4-\sqrt{5})(4+\sqrt{5})}$

Using identity $a^2-b^2=(a+b)(a-b)$

= $\frac{12+3\sqrt{5}}{{(4)}^2-{(\sqrt{5})}^2}$

Now,

= $\frac{12+3\sqrt{5}}{16-5}$

Hence, $\frac{12+3\sqrt{5}}{11}$ is the simplified equation