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!

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 / √24*√24

Simplify the top and bottom like so:

√192 / 24

Now, we need to simplify the numerator!  It turns out that 192 = 64*3, so:

√192 / 24 = √64*√3 / 24

And, this simplifies to:

8*√3 / 24

Now, divide an 8 out of top and bottom to get:

√3 / 3

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]