Simplifying Radicals “Roots” (or “radicals”) are the “opposite” operation of applying exponents. For example: If you square 2, you get 4. The square root of 4 (√4) is 2. It is a good idea to know the beginning sequence of perfect squares. This will only help you. 1 4 9 16 25 36 49 64 81 100 121 144

To simplify a radical you “take out” anything that is a perfect square. You put this perfect square root, in front of the radical sign (the radicand) √49= √7*7 = √72=7

What if you can’t see a perfect square?? Example: √75 ??? Can you see that in √75 you can factor the 75? √75=√3*25 Wait!! Is 25 a perfect square?? Why, yes it is, so if you bring the square root of 25 out front you get… 5√3 with the 3 remaining inside the radicand

Now you try some √72 = ?? √4500 = ?? √48 = ??

Answers √72= √2*36 = 6√2 √4500= √45*100= 10√45 (But wait! Can 45 be factored with a perfect square??) Yes it can!10√9*5, so the answer is 10*3√5 = 30√5 √48= √4*12 = 2√12= 2√4*3= 2*2√3= 4√3 (Or: √48=√3*16= 4√3) Are you getting the idea?? When you bring something out of the radicand you are multiplying it by what is already in front. So, even if you don’t see the perfect square right away, by using factoring you will get there eventually. You just keep going until there are no more perfect squares inside the radicand

Adding/Subtracting Radicals   Just as "you can't add apples and oranges", so also you cannot combine "unlike" radicals. To add radical terms together, they have to have the same radical part.

Example: 3√3+2√3 Because the radicals are the same ( the square root of 3) I can combine them. (This is just like adding like terms: 2x + 3x = 5x) So: 3√3+2√3=5√3 If there is no number before the radical sign, is it “understood” to be 1.

Try a few 7√5+3√5-√5= ? √13-5√13= ? 12√7-8√7 =? 7√5+3√5-√5= ? √13-5√13= ? 12√7-8√7 =? Check your answers on the next slide

Answers 9√5 -4√13 4√7