Simplifying Radicals

Q:  Simplify:

A:  There are a few ways to do this, but first I will rationalize the denominator (that means get rid of the square root in the denominator) and then I will simplify the square root (simplest radical form):

Step 1:  Rationalize the denominator:

Multiply the top and bottom by sqrt(75):

Now, simplify algebraically:

Simplify the denominator to get 150… for some reason, I can’t make my picture keep it in a fully fraction with 150 in the bottom, so the image shows the 150 pulled out like so (either way is fine):

Now reduce the 15 and the 150 to get:

The denominator has been rationalized (no square roots in the denominator) and the fraction has been reduced.

Step 2:  Put the numerator into simplest radical form:

To go to simplest radical form, you want to see what “perfect squares” divide into the square root.  So, what perfect squares divide into 75?  25 is a perfect square that divides into 75:  3*25=75.

So, we can break up the square root like so (again, the square roots can be on top of the fraction, just having image problems):

Now, what is the square root of 25?  That’s 5, so simplify:

And, now reduce the fraction part of the 5 and the 10 in the denominator to get:

You are done!

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