Ch. 8 Radical Expressions Radical Expressions are expressions with square roots in them. The square root sign (√ˉ) is also called the “radical sign.” The number or variable expression under the radical sign is called the “radicand.” The principal square root of a number is a positive number that when squared becomes the radicand. Because 72 = 49. -7 is also a square root of 49 because (-7)2 = 49. But since -7 is not positive, it is not the principal square root. To indicate the negative square root you have to attach a negative sign to the radical sign. Simplifying Square Roots Radical Expressions are in simplest form when the radicand contains no factors that are perfect squares. The Product Property of Square Roots says you can split a radical expression into its factors. Example 2: Simplify Are there any factors of 18 that are perfect squares?

Another definition of a square root is the power ½ . Taking a number or variable expression to the power ½ is the same as taking the square root. IMPORTANT!!!!!! WRONG!!

What if the exponent isn’t even What if the exponent isn’t even? How do you take the ½ th power of an odd exponent? Factor the expression into factors with even powers and factors with odd powers. Example Simplify: Simplify: First factor out any perfect squares of the coefficent, and then factor out even powers of the variables.

“the nth root of a to the nth power” Higher Roots “the nth root of a to the nth power” “Perfect Powers” a a2 a3 a4 a5 1 2 4 8 16 32 3 9 27 81 243 64 256 1,024 5 25 125 625 3,125 6 36 216 1,296 7,776 7 49 343 2,401 16,807 512 4,096 32,768

APPLICATIONS: Pythagorean Theorem C (hypotenuse) A (leg) B (leg)

Addition and Subtracting Radical Expressions Adding and subtracting radical expressions is like combining like terms. You cannot add radical expressions that have different terms inside the radical sign. Therefore, if terms have different numbers inside the radical sign and these radical expressions cannot be simplified any more, then you cannot combine them. You can, however, use the distributive property to factor out any like terms. Simplify: Simplify: