# How do you simplify 5/sqrt3 - 2/(3+sqrt3) by rationalizing the denominator?

Apr 9, 2015

Here, the denominators of both the terms need to be rationalized. So let's rationalize them one by one.

First color(red)(5/sqrt3
To rationalize the denominator, we multiply the numerator as well as the denominator by $\sqrt{3}$

$\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

= $\frac{5 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}$

= color(red)((5*sqrt3)/3(As the denominator is a Rational Number, we have successfully simplified the first term)

The second term is color(blue)(2/(3+sqrt3)

To rationalize the denominator, we multiply the numerator as well as the denominator by its Conjugate : $3 - \sqrt{3}$

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

= $\frac{2 \cdot \left(3 - \sqrt{3}\right)}{{3}^{2} - {\left(\sqrt{3}\right)}^{2}}$
We used the Identity $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$ in the Denominator

= $\frac{2 \cdot \left(3 - \sqrt{3}\right)}{9 - 3}$

= $\frac{2 \cdot \left(3 - \sqrt{3}\right)}{6}$

= color(blue)((3-sqrt3)/3

Now we can simplify the original expression $\frac{5}{\sqrt{3}} - \frac{2}{3 + \sqrt{3}}$

= color(red)((5*sqrt3)/3 $-$color(blue)((3-sqrt3)/3

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

=color(green)( ((4*sqrt3) - 3 ) / 3