Simplifying Radicals

Q:  Simplify:  sqrt(9) / sqrt(18)


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

So, I am going to break sqrt(18) = sqrt(2)*sqrt(9) since our problem already has a sqrt(9) in it.

sqrt(9) / sqrt(18) = sqrt(9) / [sqrt(2)*sqrt(9)]

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

1 / sqrt(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.

To do that in this case, you multiply the top and bottom by the denominator.  So, multiply top and bottom by sqrt(2):

sqrt(2)*1 / sqrt(2)*sqrt*(2)

And, the bottom simplifies since sqrt(2)*sqrt(2) = 2.. SO, you get:

sqrt(2)/2  [final answer]

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