1. Radicals
Simplifying

2. What is a radical?
A radical is any expression containing the radical (√) symbol. It can be known as the “root system” (square root, third root, fourth root, etc…). √: square root (what number can go in twice). √36 = 6…(6 ∙ 6 = 36) 3√: third root (what number can go in three times). 3√64 = 4…(4 ∙ 4 ∙ 4 = 64) 4√: fourth root (what number can go in four times). 4√16 = 2…(2 ∙ 2 ∙ 2 ∙ 2 = 16) There are specific rules that govern how we simplify radicals. The best thing to remember is FACTOR!

3. Simplifying Numbers in Radicals
Whenever a radical contains a number, it is best to factor that number into other numbers you know the root for. For example… What is the √25? 5. What is the √45? I don’t know. However, we can say that 45 is the same as 5 ∙ 9; therefore, √45 = √(5 ∙ 9). We may not know the √5, but the √9 = 3. So, √45 = √(5 ∙ 9) = 3√5 When you simplify and you get multiple answers out of the radicals, multiply them together. For example… What is the √80? I don’t know. However, we can say √80 = √(4 ∙ 20) = √(4 ∙ 4 ∙ 5). The √4 = 2, but there are two √4. Therefore, √80 = 2 ∙ 2√5 = 4√5.

4. Practice
5. √ √72 6√2 7. √192 8√3 8. √15 √15
Simplify the following radicals. 1. √ √ √ √49 7

5. Simplifying Variables in Radicals
Whenever a radical contains a variable, divide the exponent above the variable by the root of the radical. For example… What is the √x2? x What is the √y8? y4 What is the 3√z15? z5

6. Practice
5. 4√x20 x5 6. 3√x21 x7 7. √x4y5z6 x2y2z3√y 8. √36x8y4z 6x4y2√z
Simplify the following radicals. 1. √x3 x√x 2. √x16 x8 3. √x22 x √x15 x5