"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.You are watching: How to rationalize a denominator with two terms**

### 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**.

See more: What Does The Suffix Ment Mean, Ment Meaning

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

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:

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

(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.

### Useful

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