*
"Rationalizing the denominator" is when we move a root (like a square root or cuberoot) 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.

You are watching: How to move denominator to numerator


Example: 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 we do it?

1. Multiply Both Top and Bottom by a Root

Sometimes we 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:

*

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

See more: How To Download Rosters On Madden 21, How To Use Custom Rosters In Madden 22


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

Example Expressitop top Its Conjugate
x2 − 3 x2 + 3
Another Example Its Conjugate
a + b3 a − b3

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

(a+b)(a−b) = a2 − b2

Here is how to do it:


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:

13−√2 × 3+√23+√2 = 3+√232−(√2)2 = 3+√27

(Did you see that we used (a+b)(a−b) = a2 − b2 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!