Rationalizing the denominator is getting either an i or a square root out from the bottom of a fraction. Some problems that say to rationalize the denominator could look something like the following.

 

 

 

These type of problems will almost always say to rationalize or simplify the expression. Also, keep in mind that if there is an answer with an i or a radical in the bottom it needs to be rationalized (i.e. get the i or square root out of the bottom).

 

There are two main types of these problems, those with one term in the denominator (bottom).

 

 

and those with more than one term (almost always 2) in the bottom.

 

 

 

The easiest ones to work are those with one term in the bottom. Determine if it is an i or a square root on bottom. If it is an i, multiply it by . Here is an example.

 

 

 

 

 

Example

Simplify

 

Realize that it has an i in bottom, so multiply by .

=

 

 

From the section on imaginary numbers, i² = -1.

 

Anything divided by 1 is itself, All the negative on bottom does is change the sign.

- 2 i

 

Answer!

 

 

 

 

If there is a square root on bottom, multiply by that square root over itself. In other words, if is on bottom, then multiply by .

 

Example

Simplify

 

Realize that the bottom has a square root so multiply by that square root over itself.

 

=

 

Remember that the square root of anything times itself is that number ( = x) or anything else for x, in this case it is 6(in the bottom).

 

Simplify.

 

Thecannot simplify because it is inside a radical and there is not a perfect square that goes into it. The 4 on top and the 6 on bottom will simplify.

 

 

Answer!

 

 

 

 

 

Here are a couple more examples of problems with one term on bottom.

 

Example

Simplify

 

Notice the on bottom.

 

Multiply by .

 

 

Multiplying the top gets x. On the bottom, the result is 5(3) or 15.

 

Answer!

 

Note- In the previous answer, the 3 and 15 cannot simplify since the 3 is underneath the radical.

 

 

 

 

 

 

 

Example

Simplify

 

Recognize the i on bottom and multiply by .

=

 

 

Now i² = -1.

 

 

There is a –1 on top and one on bottom, with all the terms on top being multiplied. Cancel out each -1.

 

3

 

Answer!

 

 

 

 

 

 

 

Example

Simplify

 

Even though this denominator has a 2 being multiplied with it, multiply by .

 

Make sure both terms on top are being multiplied by i.

 

 

Remember that i² = -1.

 

 

Simplify.

 

Answer!

 

 

 

All of the previous examples had one term in the denominator, look at some of those that have more than one (almost always 2).

 

In order to do these types of problems, it is important to know what a conjugate is. If a number is written in the form of then the conjugate would be . Here is a list of some numbers and their conjugates.

 

Number	Conjugate

 

After seeing these examples it may be easier to understand the rule of conjugates as being the exact same thing being started with only the sign in the middle changed.

 

There is another kind of conjugates and they are complex conjugates. This means that they have imaginary numbers in them (in other words they have an i in them). Here are a few of those.

 

Number	Conjugate

 

 

The reason to use conjugates is to get rid of radicals (almost always on the bottom). If there is a denominator (number on bottom), multiply it by its conjugate or complex conjugate and that will eliminate the radical.

Take any number, use the first example of , and multiply by its conjugate, , here is what happens.

 

()()

Remember the rule on multiplying polynomials- Multiply every term in one by every term in the other. That means to take the 4 in and multiply it by the other 4 and the . Also take the and multiply it by 4 and .

 

4(4) = 16, 4()= - 4

(4) = + 4, and

(- )= -3

 

16 - 4 + 4 -3

 

Combining the like terms results in 13.

 

13

 

Answer!

 

Notice that the radicals (square root signs) become opposites. In other words one was positive and one was negative with the same value, in this case the value was 4. Any time one combines like terms and get a situation like that whether it is with regular numbers, radicals variables, etc… they will add up to 0, or more easily understood, they cancel out.

 

Here is an example of multiplying two complex conjugates.

 

(3 - 2i )(3 + 2i )

 

Multiply every term in one with every term in the other.

 

3(3)=9, 3(+2i ) = 6i ,

-2i (3)= - 6i , and

(-2 i )(+2i )= -4 i²x.

 

 

9 + 6i - 6i - 4 i²x

 

Make sure to multiply terms with radicals in them. Multiply everything on the outside and then everything on the inside.

 

9 + 4x

 

Combing the two like terms that cancel out (+ 6i and - 6i). Also,

i² = -1. Therefore –4(-1)x = 4x.

 

In the previous two examples it was shown how the conjugate will always cancel out the radical. Before applying this idea understand that not everything that is an expression, any math statement without an equals sign, can be multiplied by another. What has to be done instead of multiplying by the conjugate, is to multiply by the conjugate over the conjugate. The reason to do this is because anything divided by itself is always 1. Since this is true, it can only be multiplied by 1. All that is being done is changing the way an expression looks, not changing the actual value of the expression. That may seem somewhat complicated but re-read it if not fully understood the first time. It is a tough concept to grasp but very important.

 

Note- Whenever working these types of problems, it may say to simplify or rationalize. Either way, do the same thing.

 

 

 

 

 

 

Example

Simplify

 

The first thing to do is to find the conjugate of the denominator. In this case the conjugate of 2 + is 2 - . Then multiply the entire expression by the conjugate over the conjugate.

 

 

Multiplying the top terms (3 and 2 - ) produces 6 - 3. When multiplying the bottom terms (2 +and 2 - ) gets 4 - 2+ 2 - 5. On the bottom terms simplify the 4 - 2+ 2 - 5, the middle terms (-2and + 2) cancel out. That leaves 4 –5.Which is –1.

 

 

This is usually as far as one would go with rationalizing a denominator but in this case it can be simplified even more. To show this next step the first thing to do is to separate the expression.

 

-

 

Dividing anything by 1 will result in the same number started with. In this case it is being divided by –1. All that will do is change the sign of each term.

 

-6 + 3

 

Answer!

 

Note- The two negatives on the second term when divided, will result in a positive.

 

 

 

 

 

 

Here is one with the complex conjugates (it has an i in it).

 

Example

Rationalize

 

Find the complex conjugate and multiply the expression by the complex conjugate over itself.

 

 

Multiplication is understood when it looks like this. On the top, after multiplying results in 8 + 12i. The bottom produces 4 + 6i- 6i- 9i²(7). After canceling out the middle terms, results in 4 - 9i²(7).

 

 

 

When simplifying the denominator recognize that i²= - 1. Thus producing 4 – 9(-1)7. –9(-1) = 9.

 

The next step is to take that (9) and multiply it by 7. That would leave 4 + 63 = 67.

 

 

Answer!