# How do you simplify 3/(4+4sqrt5)?

Jul 3, 2017

See a solution process below:

#### Explanation:

First, factor the denominator and rewrite the expression as:

$\frac{3}{\left(4 \cdot 1\right) + \left(4 \cdot \sqrt{5}\right)} \implies \frac{3}{4 \left(1 + \sqrt{5}\right)} \implies$

$\frac{3}{4} \times \frac{1}{1 + \sqrt{5}}$

Next, multiply this expression by $\frac{1 - \sqrt{5}}{1 - \sqrt{5}}$ to eliminate the radical in the denominator while keeping the value of the expression the same because we are multiplying it by a form of $1$:

$\frac{3}{4} \times \frac{1}{1 + \sqrt{5}} \times \frac{1 - \sqrt{5}}{1 - \sqrt{5}} \implies$

$\frac{3}{4} \times \left(\frac{1}{1 + \sqrt{5}} \times \frac{1 - \sqrt{5}}{1 - \sqrt{5}}\right) \implies$

$\frac{3}{4} \times \frac{1 - \sqrt{5}}{1 - \sqrt{5} + \sqrt{5} - {\left(\sqrt{5}\right)}^{2}} \implies$

$\frac{3}{4} \times \frac{1 - \sqrt{5}}{1 - 0 - 5} \implies$

$\frac{3}{4} \times \frac{1 - \sqrt{5}}{-} 4 \implies$

$\frac{3}{4} \times - \frac{1 - \sqrt{5}}{4}$

We can now multiply the two terms giving:

$- \frac{3 \left(1 - \sqrt{5}\right)}{16}$

Jul 3, 2017

Multiply both the numerator and denominator by the conjugate of 4 + 4$\sqrt{5}$, which is 4 - 4$\sqrt{5}$. Then simplify.

#### Explanation:

$\frac{3}{4 + 4 \sqrt{5}}$ * $\frac{4 - 4 \sqrt{5}}{4 - 4 \sqrt{5}}$

Multiply these fractions together. In the denominator, you can see $\left(a + b\right) \left(a - b\right)$, which becomes ${a}^{2} - {b}^{2}$.

= $\frac{12 - 12 \sqrt{5}}{16 - 80}$

= $\frac{12 - 12 \sqrt{5}}{- 64}$

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