# How do you rationalize the denominator and simplify sqrt33 /sqrt77?

Apr 13, 2015

Before we rationalise the denominator, let's simplify the fraction.

$\frac{\sqrt{33}}{\sqrt{77}}$

$= \frac{\sqrt{3 \cdot 11}}{\sqrt{7 \cdot 11}}$

$= \frac{\sqrt{3} \cdot \cancel{\sqrt{11}}}{\sqrt{7} \cdot \cancel{\sqrt{11}}}$

$= \frac{\sqrt{3}}{\sqrt{7}}$

Now we can rationalise the denomintor by multiplying the numerator as well as the denominator by $\sqrt{7}$

 = sqrt 3 / sqrt 7 * color(blue)(sqrt 7/ sqrt 7
(We are multiplying $\frac{\sqrt{3}}{\sqrt{7}}$ with $\textcolor{b l u e}{1}$)

$= \frac{\sqrt{3} \cdot \sqrt{7}}{7}$

$= \frac{\sqrt{3 \cdot 7}}{7}$ (In general $\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$)

color(green)( = (sqrt 21) / 7

As the denominator 7 is Rational, we can say that we have Rationalised the denominator of the original fraction $\frac{\sqrt{33}}{\sqrt{77}}$