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Rational Exponents and Radicals •Definition of b1/n •Definition of bm/n •Definition of Rational Exponents n •Definition of √b n •Definition of (√b)m •Properties of Radicals •Arithmetic Operations of Radicals Definition of b1/n If n is an even positive integer and b > 0, then b1/n is the nonnegative real number such that (b1/n)n = b. If n is an odd positive integer, then b1/n is the real number such that (b1/n)n = b. If the property for multiplying exponential expressions is to hold for rational exponents, then 9½ · 9½ must equal 9½ + ½ = 91 = 9. So, 9½ must be a square root of 9. That is, 9½ = 3. 25½ = 5 (-64)⅓ = -4 Evaluate Exponential Expressions a. 16½ = √16 = 4 b. -16½ -(16½) = -√16 = -4 c. (-16)½ = √-16 not a real number d. (-32)1/5 = √-32 = (-2)5 = -2 Definition of bm/n For all positive integers m and n such that m/n is in simplest form , and for all real numbers b for which b1/n is a real number, bm/n = (b1/n)m = (bm)1/n Because bm/n is defined as (b1/n)m and also as (bm)1/n, we can evaluate 84/3 in more than one way. 84/3 = (8⅓)4 = 24 = 16 84/3 = (84)⅓ = 4096⅓ = 16 Of the two methods, (b1/n)m is usually easier to apply, provided you can evaluate b1/n Evaluate Exponential Expressions a. 8⅔ = (8⅓)² = 2² = 4 b. 324/5 = (321/5)4 = 24 = 16 c. (-9)3/2 = [(-9)½]³ (-9)½ is not a real number d. (-64)4/3 = [(-64)⅓]4 = (-4)4 = 256 Properties of Rational Exponents We looked at the properties of exponents in Section 3, but we are going to reapply them to remind you that they are extended to apply to rational exponents. Product bm · bn = bm + n Quotient bm/bn = bm – n if b ≠ 0 Power (bm)n = bmn (ambn)p = ampbmp Remember that an exponential expression is in simplest form when no power of powers or negative exponents appear. Simplify Exponential Expressions ( x²y³ ½ x-3y5 ) (x½ - y½)² (x½ - y½) (x½ - y½) (x2 – (-3)y3 – 5)½ (x5y-2)½ x5/2y-1 x5/2 y x - 2x½y½ + y Definition of √b n For all positive integers n, all integers m, and all real numbers b such that √b is a real number, (√b)m = √bm = bm/n bm/n n = √bm (5xy)2/3 (a² + 5)3/2 √(5xy)² (√a² + 5 )³ 3 √(a² + 5)³ 3 √25x²y² Evaluate Radical Expressions a. (√8)4 3 = 84/3 = (8⅓)4 = 24 = 16 b. (√9)² 4 = 92/4 = 9½ = 3 c. (√7)² = 72/2 = 71 = 7 Properties of Radicals Product √a · √b = √ab Quotient n √a n √b Index n n n √√b = √b m n mn n (√b)n = b n √bn = b A radical is in simplest form when: 1. The radicand contains only powers less than the index. 2. The index of the radical is as small as possible. 3. The denominator has been rationalized. 4. No fractions appear in the radicand. Simplify Radicals a. √32 = √25 = √2³ · √2² = 2√4 b. √12y7 = √2² · 3 · y6 · y = √2² · √3 · √y6 · √y = 2 · y³ · √3 · √y = 2y³√3y c. √162x²y5 = √2 · 34 · x² · y4 · y = √2 · √34 · √x² · √y4 · √y = 3² · x · y² · √2 · √y = 9xy√2y d. √√x8y = √x8y = √x6 · x² · y = x√x²y e. √b² = b2/4 = b½ = √b 3 3 4 3 3 6 3 6 3 6 Arithmetic Operations on Radicals Like radicals have the same radicand and the same index. Treat them the same as like terms. 3√x²y and -2√x²y 4√3 + 7√3 = (4 + 7)√3 = 11√3 Combine Radicals 5√32 + 2√128 Multiply Radical Expressions (√3 + 5)(√3 – 2) (√5x - √2y)(√5x + √2y) Rationalize the Denominator 3 √2 5 √a 3 Rationalize the Denominator 2 √3 + √a a + √5 a - √5 Rational Exponents and Radicals Assignment Page 54 – 55 # 3 – 147, multiples of 3