# How do you rationalize the denominator and simplify 5/(sqrt3-1)?

##### 1 Answer

$\frac{5 \sqrt{3} + 5}{2} = \left(\frac{5}{2}\right) \left(\sqrt{3} + 1\right)$

#### Explanation:

We can simplify this fraction by using a clever use of the number 1 and also by keeping in mind that something in the form of:

$\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$, so that will allow us to rationalize the denominator without getting into messy square root issues.

Starting with the original:

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

We can now multiply by 1 and not change the value of the fraction:

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

and we can now pick a value of 1 that will suit our purposes. We have something in the form of $\left(a - b\right)$ already in the denominator, so let's multiply by $\left(a + b\right)$:

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

which gets us:

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

simplifying:

$\frac{5 \sqrt{3} + 5}{3 - 1} = \frac{5 \sqrt{3} + 5}{2} = \left(\frac{5}{2}\right) \left(\sqrt{3} + 1\right)$