1. The problem with this country is all the unsimplified radicals running around!!

4. Most square roots are NOT exact, however. If you try to use a calculator to find their value, you get some weird decimal. No matter how many decimal places you use, however, that is still an approximation! If you are not TOLD to approximate, you should not approximate your square roots, but should SIMPLIFY them as much as possible.

5. So what does it mean to simplify a square root?? There are a couple of steps to go through: Find the largest “perfect square” that divides into your “radicand” exactly, and use it to factor your “radicand.” The radicand is the number UNDER the square root. Split your square root in two, one of which uses that perfect square you found, the second using the left- over factor. Simplify the exact square root.

6. Our calculator gives us a wacky approximation. So we look at our list of perfect squares, and find the biggest one that goes into 98 exactly: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, … Note that 49 is the biggest number on the list that goes into 98 evenly.

7. We split 98 into two factors, 49 times 2. Now, the square root of 49 is exactly 7, and therefore the expression simplifies exactly to 7 times the square root of 2.

8. We consider our list of perfect squares again, and find the largest one, if any, that divides into 108 evenly. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, … 36 is the largest one that will work, since 108 factors into 36 times 3 (that may not be obvious!).

9. We consider our list of perfect squares again, and find the largest one, if any, that divides into 108 evenly. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, … 36 is the largest one that will work, since 108 factors into 36 times 3 (that may not be obvious!).

10. We consider our list of perfect squares again, and find the largest one, if any, that divides into 65 evenly. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, … None of the perfect squares goes into 65 evenly, and so the square root of 65 does not simplify at all, and we have to leave it as it is.

11. Adding Radicals You can only add radicals if they use exactly the same square root. That is, something like this:

12. Adding Radicals You may have to simplify the radicals first!