5.6: Multiplying and Dividing Square Roots
The area of a rectangle is
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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
Vocabulary
 Rationalize the denominator
 The process used to get a radical out of the denominator of a fraction.
Practice
Simplify the following fractions.

425−−−√ 
−1649−−−√ 
96121−−−−√ 
52√10−−√ 
615−−√ 
6035−−−√ 
818−−√30−−√ 
126√ 
208143−−−−√ 
213√214−−√
Challenge Use all the Radical Rules you have learned in the last two oncepts to simplify the expressions.

812−−−√⋅15−−√ 
3245−−−√⋅620−−√5√ 
24−−√2√+826−−√8√ 
2√3√+46√3√ 
55√12−−√−215−−√10−−√
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Here you'll learn how to divide radicals and rationalize the denominator.