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Multiplication and Division of Radicals

Rationalize the denominator

Atoms Practice
Practice Multiplication and Division of Radicals
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Dividing Square Roots

The area of a rectangle is 30. The length of the rectangle is 20. What is the width of the rectangle?

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

Khan Academy: How to Rationalize a Denominator


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. ab=ab

5. abbb=abb

Example A

Simplify 14.

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


Example B

Simplify 23.

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 3.


Example C

Simplify 3240.

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.

3020 = 32.

Now we need to rationalize the denominator. Multiply the top and bottom of the fraction by 2.


Therefore, the width of the rectangle is 62.

Guided Practice

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

1. 12

2. 6450

3. 436


1. 12=12=1222=22

2. 6450=3225=1625=425

3. The only thing we can do is rationalize the denominator by multiplying the numerator and denominator by 6 and then simplify the fraction.


Explore More

Simplify the following fractions.

  1. 425
  2. 1649
  3. 96121
  4. 5210
  5. 615
  6. 6035
  7. 81830
  8. 126
  9. 208143
  10. 213214

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

  1. 81215
  2. 32456205
  3. 242+8268
  4. 23+463
  5. 551221510


Rationalize the denominator

Rationalize the denominator

To rationalize the denominator means to rewrite the fraction so that the denominator no longer contains a radical.

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