# Rationalize the Denominator

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

## Oh No! An Irrational Denominator!

The bottom of a fraction is called the **denominator**.

Numbers like 2 and 3 are rational.

But many roots, such as √2 and √3, are irrational.

### Example: \frac{1}{√2} has an Irrational Denominator

To be in "simplest form" the denominator should **not** be irrational!

Fixing it (by making the denominator rational)

is called "**Rationalizing the Denominator**"

Note: an irrational denominator is not **wrong**, it still works. But it is not "simplest form" and so **can cost you marks**.

Also equations can be **easier to solve** and **calculations can be easier** without an irrational denominator, so you should learn how.

So ... how?

## 1. Multiply Both Top and Bottom by a Root

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

### Example: \frac{1}{√2} has an Irrational Denominator. Let's fix it.

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

\frac{1}{√2} × \frac{√2}{√2} = \frac{√2}{2}

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

Having an irrational number in the top (numerator) of a fraction is preferred.

See below for a comparison between calculating \frac{1}{√2} and \frac{√2}{2}

## 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 ... we **multiply both top and bottom** by the** conjugate of the denominator**.

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

^{2}− 3

^{2}+ 3

^{3}

^{3}

The method we will see works because when we multiply something by its conjugate we get **squares** like this:

(a+b)(a−b) = a^{2} − b^{2}

Here is how to do it:

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

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

We can * multiply both top and bottom by 3+√2 (the conjugate of 3−√2)*, which won't change the value of the fraction:

\frac{1}{3−√2} × \frac{3+√2}{3+√2} = \frac{3+√2}{3^{2}−(√2)^{2}} = \frac{3+√2}{7}

(Did you see that we used **(a+b)(a−b) = a ^{2} − b^{2}** in the denominator?)

*Use your calculator to work out the value before and after ... is it the same?*

There is another example on the page Evaluating Limits (advanced topic) where I move a square root from the top to the bottom.

### Useful

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

For fun let's try calculating both \frac{1}{√2} and \frac{√2}{2}

They should be the same. Let's use 1.414 as a rough estimate for √2

For \frac{1}{√2} ≈ \frac{1}{1.414} we need long division:

1.414 )1.000

0

1.0000

0.9898

0.0102

etc... give up, too hard!

The answer is 0.70... something.

Now let's try \frac{√2}{2} ≈ \frac{1.414}{2}:

\frac{1.414}{2} = 0.707 easy!

Which form do you prefer?