Square Roots and Radicals[1]

Square Roots; Radicals In general, if a is a nonnegative real number, the nonnegative number b such that b  a is 2 the principal square root of a and is denoted by b  a . a a 2 . . . 51  51  51 2 . . .   51   51  51 2 z z 2 Product Property of Square Roots a b  ab 18  9  2  9 2  3 2 50 x  3 25x 2 x 2 2  25x 2x  5 x 2x Quotient Property of Square Roots a a  b b When radicals occur in quotients, it is common to rewrite the quotient so that the denominator contains no radicals. This process is called rationalizing the denominator. Rationalize the denominator in each expression. 3 2 3 2 3 2    2 2 2 2 a  3 a  3  a  3 a 3    a 3 a 3 a 3 a9 The principal nth root of a real number a, n symbolized by a is defined as follows: n a  b means a  b n where a > 0 and b > 0 if n is even and a, b are any real numbers if n is odd n n a  a, n n if n is odd if n is even a  a, 81  9 3 because 9  81 2 3  27  3 because - 3  27 5 5 12  12 5   12  12 5 6 6 8 5 5 6   5   5  5 6 z z 8 Properties of Radicals n ab  a b n n n n a a n b b a   a m n m n mn a mn a 4 32 x  5 4 4 16x 2 x 4 44  16 x 4 2x  2 x 2x