"Rationalizing the denominator" is when we move a source (like a square source or cuberoot) native the bottom that a portion to the top.

Oh No! one Irrational Denominator!

The bottom that a fraction is referred to as the denominator. Numbers like 2 and also 3 are rational.But many roots, such as √2 and √3, room irrational.

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Example: has actually an Irrational Denominator


To be in "simplest form" the denominator must not it is in irrational!

Fixing that (by do the denominator rational)is called "Rationalizing the Denominator"

Note: over there is nothing wrong v an irrational denominator, it still works. Yet it is no "simplest form" and so can price you marks.

And removing them may aid you resolve an equation, so you should learn how.

So ... How do we execute it?

1. Multiply Both Top and Bottom through a Root

Sometimes we can just multiply both top and also bottom through 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 reasonable number (=2). Done!

Note: that is ok to have actually an irrational number in the peak (numerator) of a fraction.

2. Main point Both Top and Bottom through the Conjugate

There is one more special means to move a square root from the bottom of a portion to the top ... We multiply both top and also bottom by the conjugate of the denominator.

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

example Expression that Conjugate
x2 − 3 x2 + 3
an additional Example that is Conjugate
a + b3 a − b3

It works since when we multiply something by its conjugate we gain squares choose this:

(a+b)(a−b) = a2 − b2

Here is just how to carry out it:

How deserve to we move the square source of 2 come the top?

We can multiply both top and also bottom by 3+√2 (the conjugate the 3−√2), i beg your pardon won"t adjust the value of the fraction:

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

(Did you view that we provided (a+b)(a−b) = a2 − b2 in the denominator?)

Use her calculator to work-related out the value before and also after ... Is that the same?

There is an additional example ~ above the page analyzing Limits (advanced topic) wherein I relocate a square root from the optimal to the bottom.


So try to psychic these little tricks, it may aid you deal with an equation one day!