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"Rationalizing the denominator" is when we relocate a source (like a square source or cuberoot) native the bottom of a fraction to the top.

Oh No! an Irrational Denominator!

The bottom that a fraction is called the denominator. Numbers choose 2 and also 3 are rational.But numerous roots, such as √2 and √3, are irrational.

You are watching: How to move denominator to numerator


Example: has an Irrational Denominator

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

Fixing that (by making the denominator rational)is referred to as "Rationalizing the Denominator"

Note: there is nothing wrong with an irrational denominator, that still works. However it is not "simplest form" and so can cost you marks.

And removing lock may aid you settle an equation, so you should find out how.

So ... Exactly how do we perform it?

1. Multiply Both Top and Bottom through a Root

Sometimes we have the right to just multiply both top and also 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:

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


Note: the is yes sir to have actually an irrational number in the height (numerator) of a fraction.

2. Multiply Both Top and Bottom by the Conjugate

There is one more special method to relocate a square root from the bottom the a fraction to the peak ... We multiply both top and also bottom by the conjugate of the denominator.

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The conjugate is where we change the authorize in the middle of 2 terms:

instance Expression that is Conjugate
x2 − 3 x2 + 3
another Example that is Conjugate
a + b3 a − b3

It works since when we multiply something by the conjugate we acquire squares favor this:

(a+b)(a−b) = a2 − b2

Here is exactly how to execute it:


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

We have the right to multiply both top and bottom by 3+√2 (the conjugate of 3−√2), which won"t change the worth of the fraction:

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

(Did you check out that we offered (a+b)(a−b) = a2 − b2 in the denominator?)

Use her calculator to job-related out the worth before and after ... Is it the same?


There is one more example on the page assessing Limits (advanced topic) where I move a square root from the height to the bottom.

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