Simplifying Radical Expressions - Problem 1

In this problem I’m asked to simplify the square root of 120. Now a lot of students when they first get to these problems are going to grab their calculators and they’re going to say wait a second there is no square root of 120. Well you’re kind of right, the square root of 120 isn’t a whole number but if you try it on your calculator, you’ll see the square root of 120 is the decimal 10.94 blah, blah, blah. 10 point 94, that’s my, oh 95 sorry that’s my decimal approximation. To show it’s an approximation I would write instead of an equal sign, I’d write two little squiggles, but that’s not the answer we’re looking for. The whole point of simplifying square roots is to be able to rewrite this using perfect square factors, so let me think about this.

When approaching problems where I’m asked to simplify square roots, I want to look for factors that are perfect squares meaning of my list of perfect squares like 1, 4, 9, 16 blah, blah, blah I’m looking for things that multiply into 120. Right away, I personally thought that 4 multiplies into 120. The square root of 120 is equal to the square root of 4 times the square root of 30. I know what the square root of 4. The square root of 4 is 2, so I’m working with so far 2 times the square root of 30.

Now what I want to do is look at 30 see if I can think of any perfect square numbers that multiply into 30. Again my perfect square numbers are 1 which I know goes into 30 because 1 goes into everything, so that can help me 1, 4, 4 doesn’t go into 30, 9 doesn’t go into 30, what’s next 1, 2, 3 16 doesn’t go into 30 nothing else goes into 30, so I think that this is going to be my final.

I think the simplified form of square root of 120 is 2 times the square root of 30. We don’t write it without multiply sign there; we just write it like that. That’s my final answer. The way I could check is by doing on my calculator 2 times the square root of 30 and see if I get 10.95 and indeed I do.

The thing that’s really interesting or really important to know about these problems is that 10.95 again is an approximation. The real answer is like 10 point 9543 blah, blah, blah it’s something really long, that’s an approximation it’s not exact. This is an exact simplification of square root of 120. That’s why it’s an important skill to have. You want to make sure you’re able to keep the exact numbers especially when you get into your careers of Math and Science; you want to have exact numbers rather than decimal approximations.