How do you simplify the radical expression: #sqrt 120 / sqrt 5#?

1 Answer

Cancel out what is common in the numerator and denominator, then simplify the result and get to


Questions like this one require that we find something in the numerator and something in the denominator that are the same, then we cancel them out and get a simplified expression.

Before we dive in, let's look at the fraction. Let's ignore the square roots for the moment! In the denominator there is a 5. In the numerator there is a 120. Can we evenly divide 5 into 120? Yes - 5 goes into 120 24 times. So if there wasn't the square roots, things would look like this:

#120/5 = (24*5)/5 = 24#

With both the numerator and denominator having square roots, the above will work the same way:

#sqrt(120)/sqrt(5) = sqrt(24*5)/sqrt(5) = (sqrt(24)*sqrt(5))/sqrt(5) = sqrt(24)#

Now we can simplify #sqrt(24)#. We do that by finding a perfect square as a factor of the number in the square root (if there isn't one, we're done). Is there one? Yes! 4 is a perfect square.

#sqrt(24) = sqrt(4*6) = sqrt(4)*sqrt(6) = 2sqrt(6)#