SECTION A.1 Algebra Essentials A9 7 Evaluate Square Roots A real number is squared when it is raised to the power 2. The inverse of squaring is finding a square root . For example, since 6 36 2 = and 6 36, 2 ( ) − = the numbers 6 and 6− are square roots of 36. The symbol , called a radical sign , is used to denote the principal , or nonnegative, square root. For example, 36 6. = NOTE Always write the final answer using positive exponents. j In Words 36 means “what is the nonnegative number whose square is 36?” DEFINITION Principal Square Root If a is a nonnegative real number, the nonnegative number b for which b a 2 = is the principal square root of a , and is denoted by b a. = Evaluating Square Roots (a) 64 8 = (b) 1 16 1 4 = (c) 1.4 1.4 2 ( ) = EXAMPLE 12 Using the Laws of Exponents Write each expression so that all exponents are positive. (a) x y x y x y 0, 0 5 2 3 ≠ ≠ − (b) x y x y 3 0, 0 3 1 2 ≠ ≠ − − − Solution EXAMPLE 11 (a) x y x y x x y y x y x y x y x y 1 5 2 3 5 3 2 5 3 2 1 2 3 2 3 2 3 = ⋅ = ⋅ = = ⋅ = − − − − − − (b) x y x y x y x y x y 3 3 3 1 9 9 3 1 2 3 2 1 2 6 2 1 2 6 2 6 2 ( ) ( ) ( ) = = = = − − − − − − − − − − Now Work problems 89 and 99 The following comments are noteworthy: • Negative numbers do not have square roots (in the real number system), because the square of any real number is nonnegative. For example, 4− is not a real number, because there is no real number whose square is 4. − • The principal square root of 0 is 0, since 0 0. 2 = That is, 0 0. = • The principal square root of a positive number is positive. • If c 0, ≥ then c c. 2 ( ) = For example, 2 2 2 ( ) = and 3 3. 2 ( ) = Examples 12(a) and (b) are examples of square roots of perfect squares, since 64 82 = and 1 16 1 4 . 2 ( ) = Consider the expression a . 2 Since a 0, 2 ≥ the principal square root of a2 is defined whether a 0 > or a 0. < However, since the principal square root is nonnegative, we need an absolute value to ensure the nonnegative result. That is, a a a any real number 2 = (2)
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