"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:

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:

*1***3−√2** × *3+√2***3+√2** = *3+√2***32−(√2)2** = *3+√2***7**

(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!