Rationalize the Denominator

What is it?

"Rationalising the denominator" is when you move a root (like a square root or cube root) from the bottom of a fraction to the top.

Why is it called "Rationalizing the Denominator" ?

The bottom of a fraction is called the denominator, and many roots are irrational, so (for example) this:

has an "Irrational Denominator" (√2 is irrational).

To be in "simplest form" you shouldn't have an irrational number in the denominator!

So fixing it (making the denominator rational) is called "Rationalizing the Denominator"

So ... how do you do it?

1. Multiply Both Top and Bottom by a Root

Sometimes you can just multiply both top and bottom by a root:

Example: has an Irrational Denominator. Let's fix it.

Multiply top and bottom by the square root of 2, because: √2 × √2 = 2:

rationalized

Now the denominator has a rational number (=2). Done!

Note: It is ok to have an irrational number in the top (numerator) of a fraction.

2. Multiply Both Top and Bottom by the Conjugate

There is another special way to move a square root from the bottom of a fraction to the top ... you multiply both top and bottom by the conjugate of the denominator.

The conjugate is where you change the sign in the middle of two terms:

Example Expression	Its Conjugate
x² - 3	x² + 3
a + b³	a - b³

Here is how you do it:

Example: here is a fraction with an "irrational denominator":

How can we move the square root of 2 to the top?

Answer! Multiply both top and bottom by the conjugate (this will not change the value of the fraction), like this:

So try to remember these little tricks, it may help you solve an equation one day!