83 R.8 Radical Expressions We use the familiar notation 2a instead of 22 a for the square root. For even values of n (square roots, fourth roots, and so on), when a is positive, there are two nth roots, one positive and one negative. !n a represents the positive root, the principal nth root. −!n a represents the negative root. Examples: 2144 = 12 and -2144 = -12 Radical Notation for am/n Let a be a real number, m be an integer, n be a positive integer, and 2n a be a real number. am/n =A!n aBm=!n am EXAMPLE 1 Evaluating Roots Write each root using exponents and evaluate. (a) 24 16 (b) -24 16 (c) 25 -32 (d) 23 1000 (e) B6 64 729 (f) 24 -16 SOLUTION (a) 24 16 = 161/4 = 2 (b) -24 16 = -161/4 = -2 (c) 25 -32 = 1-3221/5 = -2 (d) 23 1000 = 10001/3 = 10 (e) B6 64 729 = a 64 729b 1/6 = 2 3 (f) 24 -16 is not a real number. S Now Try Exercises 11, 13, 15, and 17. EXAMPLE 2 Converting from Rational Exponents to Radicals Write in radical form and evaluate if possible. Assume all variable expressions represent positive real numbers. (a) 82/3 (b) 1-3224/5 (c) -163/4 (d) x5/6 (e) 3x2/3 (f) 2p1/2 (g) 13a + b21/4 SOLUTION (a) 82/3 = A 23 8 B 2 = 22 = 4 (b) 1-3224/5 = A 25 -32 B 4 = 1-224 = 16 (c) -163/4 = -A 24 16 B 3 = -1223 = -8 (d) x5/6 = 26 x5 (e) 3x2/3 = 323 x2 (f) 2p1/2 = 22p (g) 13a + b21/4 = 24 3a + b S Now Try Exercises 23, 25, and 29.
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