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Radicals in Fractions - Rationalizing Denominators

Although sometimes, when we do a math problem, we end up with a radical in the denominator a fraction, we're not allowed to leave that radical in the denominator.

Why not?

Technically, fractions are supposed to have integers in the numerator and denominator. But mostly, it's customary to rationalize the denominator of fractions, in other words, in order for most math teacher to count an answer as "correct" there cannot be a radical in the denominator (although, interestingly, a radical in the numerator is allowed). 

How do you get rid of a radical?  Remember, a square root squared is the number inside the square root:

$\sqrt{3}^2=\sqrt{3}\times \sqrt{3}=3$

The same is true for other roots. In order to turn to get ride of a cube root, cube the term.  In order to get rid of a root to the power of 4, raise the term to the power of 4.  

$\sqrt[3]{4}^3=\sqrt[3]{4}\times \sqrt[3]{4} \times \sqrt[3]{4}=4$

So, if you have a radical in the denominator, you have to raise that term to the power of the radical (and remember, to change the denominator of a fraction, you have to multiply to a fraction that is equal to one, so you have to multiple the numerator too).

$\dfrac{2}{\sqrt{3}} = \dfrac{2}{\sqrt{3}}\times\dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{2\sqrt{3}}{3}$

And, there you go, the denominator is rationalized and this answer would be accepted on a test!

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