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College Prep 8.1 Radical Expressions and Graphs Square Root: The number c is a square root of a if c 2 a. 2 9 has –3 and 3 as square roots because 3 9 and 32 9. The principal square root of a nonnegative number is its nonnegative square root. Whenever we talk about “the square root” of a number, we mean its principal square root. The square root of a negative number is not a real number. The symbol is called a radical sign and is used to indicate the principal square root of the number over which it appears. The number under the radical sign is called the radicand. If a is a nonnegative number, then of a. a is the principal square root of a, and a is the negative square root Perfect squares are numbers that are the squares of rational numbers. Examples: 1, 4, 9, 81, Examples: Simplify each of the following: 36 a) 121 b) 49 c) 81 1 36 , 16 25 , etc. 0.36 d) Simplifying a 2 : For any real number a, a 2 a . Remember: The principal square root must be positive. When a radicand is the square of a variable expression, 2 like x 5 or 36t 2 , we must use absolute-value signs unless we know that the expression being squared is nonnegative. Examples: Simplify each expression. Assume that the variable can represent any real number. x 5 a) 2 x2 6 x 9 b) c) y4 t10 d) Examples: Simplify each expression. Assume that no radicands were formed by squaring negative quantities. y6 a) b) 4 x2 12 x 9 Cube Root: The number b is the cube root of a if and only if b3 a. 3 a denotes the cube root of a. Examples: Simplify each expression a) 3 27 b) 3 27 c) 3 27 d) 3 8m3 Odd and Even nth roots In the expression n a , n is called the index and we are finding the nth root of a. If the index is even, such as If the index is odd, such as 3 or 4 , the radicand must be nonnegative for the root to be a real number. or 5 , the radicand can be either positive or negative. Simplifying nth roots n n a a Positive Positive Negative Not a real number Positive Positive Negative Negative Even n an a a Odd Examples: Simplify each expression, if possible. Assume that variables can represent any real number. a) 5 243 b) 5 243 c) 5 243 d) 5 243 e) 9 x9 i) 4 16 f) 7 2 x 3 j) 4 16x 4 7 g) 4 16 h) x8 l) k) 4 16 4 6 m 3 6 Domain: The domain of an even root is all values of the variable that make the radicand nonnegative. The domain of an odd root is . Examples: Determine the domain of each function described. a) f x x 3 b) 3 y 7 c) g x 4 2 x 5 d) 7 3 2z Examples: Graph each function by creating a table of values. Give the domain and range. a) f x x 1 b) g x 3 x 4 HW 8.1 Problems: 13,15,18,21,24,27,30,32,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78