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Roots of Real Numbers and Radical Expressions Definition of th n Root For any real numbers a and b and any positive integers n, if b^n = a, then b is the nth root of a. ** For a square root the value of n is 2. Notation radical index 4 81 radicand Note: An index of 2 is understood but not written in a square root sign. 4 Simplify 81 To simplify means to find x in the equation: 4 x = 81 Solution: 4 81= 3 Principal Root The nonnegative root of a number 64 Principal square root 64 Opposite of principal square root 64 Both square roots Summary of Roots a The n th root of a n a>0 a<0 a=0 n even odd one + root one - root one + root no - roots no real roots no + roots one - root one real root, 0 Rational Exponents In other words, exponents that are fractions. Definition of a 1 n For any real number a and any integer n > 1, 1 n a a except when a < 0 and n is even n Examples: 1 36 6 1. 36 2 1 2. 64 3 3 64 4 Examples: 1 1 2 2 3. 49 49 1 1 1 49 7 1 1 4. 83 8 3 1 3 82 Definition of Rational Exponents For any nonzero number b and any integers m and n with n > 1, a except when b < 0 and n is even m n a a n m n m NOTE: There are 3 different ways to write a rational exponent 4 27 27 3 3 4 3 4 27 Examples: 36 6 216 1. 36 27 3 81 2. 27 81 3 27 3. 81 3 3 2 4 3 3 3 4 4 3 4 4 3 3