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

Aug 10, 2017

To rationalise this denominator, you have to take the denominator's conjugate. The conjugate will evaluate into a difference of two squares ( ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$ ) which can be expressed as an integer.

So to rationalise this expression, $\frac{5}{\sqrt{3} - 1}$

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

= $\frac{5}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1}$

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

= $\frac{5 \sqrt{3} + 5}{2}$

Aug 10, 2017

color(green)((5(sqrt3+1))/2

#### Explanation:

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

$\sqrt{3} \times \sqrt{3} = 3$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\sqrt{3} - 1$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$\times \underline{\sqrt{3} + 1}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$3 - \sqrt{3}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$$\sqrt{3} - 1$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\overline{3 + 0 - 1}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$color(green)(=2

:.color(green)((5(sqrt3+1))/2

~~~~~~~~~~~~~~~~~~~~~

check by calculator:

:.5/(sqrt3-1)=color(green)(6.830127019

:.(5(sqrt3+1))/2=color(green)(6.830127019