Rational Exponents and Radicals Definition of b 1/n Definition of b m/n Definition of Rational Exponents Definition of √b Definition of (√b) m Properties of Radicals Arithmetic Operations of Radicals n n

Definition of b 1/n If n is an even positive integer and b > 0, then b 1/n is the nonnegative real number such that (b 1/n ) n = b. If n is an odd positive integer, then b 1/n is the real number such that (b 1/n ) n = b. If the property for multiplying exponential expressions is to hold for rational exponents, then 9 ½ · 9 ½ must equal 9 ½ + ½ = 9 1 = 9. So, 9 ½ must be a square root of 9. That is, 9 ½ = ½ = 5(-64) ⅓ = -4

Evaluate Exponential Expressions a.16 ½ b.-16 ½ c.(-16) ½ d.(-32) 1/5 = √16 = 4 -(16 ½ ) = -√16 = -4 = √-16not a real number = √-32 = (-2) 5 = -2

Definition of b m/n For all positive integers m and n such that m/n is in simplest form, and for all real numbers b for which b 1/n is a real number, b m/n = (b 1/n ) m = (b m ) 1/n Because b m/n is defined as (b 1/n ) m and also as (b m ) 1/n, we can evaluate 8 4/3 in more than one way. 8 4/3 = (8 ⅓ ) 4 = 2 4 = /3 = (8 4 ) ⅓ = 4096 ⅓ = 16 Of the two methods, (b 1/n ) m is usually easier to apply, provided you can evaluate b 1/n

Evaluate Exponential Expressions a.8 ⅔ b.32 4/5 c.(-9) 3/2 d.(-64) 4/3 = (8 ⅓ )² = 2² = 4 = (32 1/5 ) 4 = 2 4 = 16 = [(-9) ½ ]³(-9) ½ is not a real number = [(-64) ⅓ ] 4 = (-4) 4 = 256

Properties of Rational Exponents Productb m · b n = b m + n Quotientb m /b n = b m – n if b ≠ 0 Power(b m ) n = b mn (a m b n ) p = a mp b mp 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. Remember that an exponential expression is in simplest form when no power of powers or negative exponents appear.

Simplify Exponential Expressions x²y³ x -3 y 5 ( ) ½ (x 2 – (-3) y 3 – 5 ) ½ (x 5 y -2 ) ½ x 5/2 y -1 x 5/2 y (x ½ - y ½ )² (x ½ - y ½ ) x - 2x ½ y ½ + 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 = √b m = b m/n b m/n = √b m n (5xy) 2/3 √(5xy)² 3 √25x²y² 3 (a² + 5) 3/2 (√a² + 5 )³ √(a² + 5)³

Evaluate Radical Expressions a.(√8) 4 b.(√9)² c.(√7)² 3 4 = 8 4/3 = (8 ⅓ ) 4 = 2 4 = 16 = 9 2/4 = 9 ½ = 3 = 7 2/2 = 7 1 = 7

Properties of Radicals Product√a · √b = √ab Quotient√a √b Index √ √b = √b (√b) n = b √b n = b n nn n n n m mn n n 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 b.√12y 7 c.√162x²y 5 d.√ √x 8 y e.√b² = √2 5 = √2³ · √2² = 2√ = √2² · 3 · y 6 · y = √2² · √3 · √y 6 · √y = 2 · y³ · √3 · √y = 2y³√3y = √2 · 3 4 · x² · y 4 · y = √2 · √3 4 · √x² · √y 4 · √y = 3² · x · y² · √2 · √y = 9xy√2y = √x 8 y = √x 6 · x² · y = x√x²y = b 2/4 = b ½ = √b

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