Rationalize the Denominator

  "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.  

Oh No! An Irrational Denominator!

The bottom of a fraction is called the denominator.
Many roots, such as √2 and √3, are irrational, but numbers like 2 and 3 are rational.

Example: 1 divide root 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: there is nothing wrong with an irrational denominator, it still works, but it is not "simplest form" and so can cost you marks. And removing them may help you solve an equation, so you should learn how.

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: 1 divide root 2 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

Another Example Its Conjugate
a + b3 a - b3

It works because when you multiply something by its conjugate you get squares like this:

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 of 3-√2 (this will not change the value of the fraction), like this:

(Did you see how the denominator became a2-b2 ?)

Useful

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

 

 
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