No wonder Pythagoras tried to hide in a cave – too much neon in this theorem

Today’s ACT question of the day is asking: do you know the Pythagorean theorem, and can you simplify a radical expression?

If you do and can, this is a 20-second question.  If you don’t, you should the Pythagorean theorem now: a^2 + b^2 = c^2.  That is, the square of the length of each side (or “leg”) of a right triangle is equal to the square of the length of the hypotenuse.  This theorem makes an appearance dozens of times in various ways throughout your test, so keep it with you.

Now that we’ve got that out of the way, let’s apply it!

We are given the length of each leg: 8 and 12.  If you know your multiplication tables up to 12, you don’t even need to break out a calculator.

8^2 = 64    12^2 = 144    64 + 144 = 208

I don’t, offhand, know the factors of 208, but I do see that most of my answer choices are 4 * the square root of something.  In order to have a 4 outside the radical, I have to have 16 underneath the radical (or 4^2, so I can take the square root of that factor and turn it into 4), so I can try dividing 208/16 to see what the other factor would be.

208/16 = 13

So, 4√13 is our answer.

Remember that to simplify a radical expression (the square root of something, or something “under the hat”), we look for a perfect square (4, 9, 16, 25, 36, 49, 64…) in the factors of whatever oddity is under the radical so that we can “pull out” the perfect square and turn it into its square root.  Whatever is left over has to stay under the radical (this thing √ ).

For example, to simplify √9, we would just take the square root of 9 and get 3.

To simplify √18, we could first look at it as √(9 * 2), pull out the 9 and take its square root, and be left with 3√2.