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Simplify by Rationalizing the Denominator

Q:  Simplify by rationalizing the denominator:  √8/√24

Answer:  To rationalize the denominator, multiply both top and bottom by the denominator.  So, multiply both top and bottom by √24:

√8/√24

√8*√24 / √24*√24

Simplify the top and bottom like so:

√192 / 24

Now, we need to simplify the numerator!  We do this by factoring the numerator into perfect squares.  It turns out that 192 = 64*3, so:

√192 / 24 = √64*√3 / 24

And, this simplifies to:

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Simplifying Radicals

Q:  Simplify: √(9) / √(18)

Answer:

First, we gotta know that we can break square roots apart into their factors.  So, √(18) can be broken up into √(3)*√(6) since 3*6 is 18… Or, √(18) can be broken up into √(2)*√(9) since 2*9 is 18.

So, I am going to break √(18) = √(2)*√(9) since our problem already has a √(9) in it (and since √(9) is a number we know!).

√(9) / √(18) = √(9) / [√(2)*√(9)]

Now, there is a √(9) on top and on bottom, so it can cancel out to leave:

1 / √(2)

However, depending on what class you are in and your teacher, you may need to rationalize the denominator.  Rationalizing the denominator means to get all square roots out of the denominator and into the numerator only.

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