Speed Bump #3 – Simplifying Radicals Before we hit our speed bump, let’s review exponents. An exponent tells you how many times a number is multiplied by itself. For example, 2! = 2×2×2 = 8 Remember that exponents ONLY apply to the term right in front of them unless there are parentheses. 3! = 3 × 3 3𝑎
!
= 3×𝑎×3×𝑎 = 9𝑎! Radicals are the inverses of exponents. The square root of 9 is 3. 9 = 3×3 = 3 since the index of a square root is 2, any number that occurs twice under the radical comes outside once. The index is the small number that sits on the ledge of the radical !
ℎ𝑎𝑠 𝑎𝑛 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 3 𝑎𝑛𝑑
!
ℎ𝑎𝑠 𝑎𝑛 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 4. Your first step when simplifying a radical is to factor the terms underneath. Then use your index to determine how you must group them in order to bring them outside the radical sign. !
16 =
!
2×2×2×
2 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 3 𝑠𝑜 𝑡ℎ𝑟𝑒𝑒 2! 𝑠 𝑐𝑜𝑚𝑒 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑎𝑠 1 𝑎𝑛𝑑 𝑦𝑜𝑢𝑟 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 2∛2 Things can get a bit more complicated. You will see problems that look like this 98𝑥 ! 𝑦 ! = 2 ∙ 7 ∙ 7 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 the index is 2 so the answer is 7𝑥 ! 𝑦 2𝑥y It is to your benefit to learn the following exponential equivalents. 2! =4 2! =8 2! =16 2! =32 2! = 64 2! =128 2! = 256 3! =9 3! =27 3! = 81 5! = 25 5! = 125 5! = 625 7! = 49 7! = 343 Practice Problems are below give them a try.