# How do you rationalize the denominator & simplify (5 + sqrt3)/(2 -sqrt3)?

Apr 16, 2015

To rationalise the denominator, we multiply the Numerator and the Denominator of this fraction with the Conjugate of the Denominator

$\frac{5 + \sqrt{3}}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$

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

The Denominator is in the form color(blue)((a-b)*(a+b) which equals color(blue)(a^2 - b^2

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

$= \frac{\left(5 + \sqrt{3}\right) \left(2 + \sqrt{3}\right)}{4 - 3}$

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

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

Using the Distributive Property of Multiplication we get:

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

$= 10 + 7 \sqrt{3} + 3$

color(green)(= 13 + 7 sqrt 3