The area of a rectangle is . The length of the rectangle is . What is the width of the rectangle?

### Watch This

Watch the first part of this video, until about 3:15.

Khan Academy: How to Rationalize a Denominator

### Guidance

Dividing radicals can be a bit more difficult that the other operations. The main complication is that you cannot leave any radicals in the denominator of a fraction. For this reason we have to do something called
**
rationalizing the denominator
**
, where you multiply the top and bottom of a fraction by the same radical that is in the denominator. This will cancel out the radicals and leave a whole number.

**
Radical Rules
**

4.

5.

#### Example A

Simplify .

**
Solution:
**
Break apart the radical by using Rule #4.

#### Example B

Simplify .

**
Solution:
**
This might look simplified, but radicals cannot be in the denominator of a fraction. This means we need to apply Rule #5 to get rid of the radical in the denominator, or rationalize the denominator. Multiply the top and bottom of the fraction by
.

#### Example C

Simplify .

**
Solution:
**
Reduce the fraction, and then apply the rules above.

**
Intro Problem Revisit
**
Recall that the area of a rectangle equals the length times the width, so to find the width, we must divide the area by the length.

= .

Now we need to rationalize the denominator. Multiply the top and bottom of the fraction by .

Therefore, the width of the rectangle is .

### Guided Practice

Simplify the following expressions using the Radical Rules learned in this concept and the previous concept.

1.

2.

3.

#### Answers

1.

2.

3. The only thing we can do is rationalize the denominator by multiplying the numerator and denominator by and then simplify the fraction.

### Vocabulary

- Rationalize the denominator
- The process used to get a radical out of the denominator of a fraction.

### Practice

Simplify the following fractions.

**
Challenge
**
Use all the Radical Rules you have learned in the last two concepts to simplify the expressions.