Information about Rational Exponents

Definition of Rational Exponents For any nonzero number b and any integers m and n with n > 1, ( ) m m n m n b = b = n b except when b < 0 and n is even

NOTE: There are 3 different ways to write a rational exponent ( ) 4 4 3 4 3 27 = 27 = 3 27

Examples: 3 ( ) 3 36 = (6) = 216 3 2= 1. 36 4 2. 27 = ( 27) = (3) = 81 4 4 3 3 3 = ( 81 = (3) = 27 ) 3 3 4 3. 814

Evaluating and Simplifying Expressions with Rational Exponents

Evaluate vs. Simplify Evaluate – finding the numerical value. Simplify – writing the expression in simplest form.

Simplifying Expressions No negative exponents No fractional exponents in the denominator No complex fractions (fraction within a fraction) The index of any remaining radical is the least possible number

In this case, we use the laws of exponents to simplify expressions with rational exponents.

Properties of Rational Exponents n Definition 1 −n 1 1 − 1 1 3 1 1 a = = n of negative 8 3 3 = = = a a exponent 8 8 2 1 1 1 =a n Definition = 36 = 36 = 6 2 of negative 1 − −n a exponent 36 2

Properties of Rational Exponents (a ) 1 Power-to- m n 1 =a mn 2 1 1 1 a-power 2 3 = ( 2) 3⋅ 2 = 2 6 Law ( ab ) m Product-t0 1 1 =a b m m -a-power ( xy ) 1 2 =x 2 y 2 Law

Properties of Rational Exponents Quotient- 1 m m to-a- 1 a a 4 2 4 2 4 2 = m power = 1 = 25 = 25 5 b b Law 25 2 −m m Quotient- 1 1 a b − to-a- 4 25 2 25 5 2 = negative- = = = b a 25 4 4 2 power Law

Exampl e: Write as a Radical and Evaluate 1 49 = 49 = 2 2 49 = 49 = 7

Example: Simplify each expression 1 1 5 2 3 5 4 ⋅a ⋅b = 4 ⋅a ⋅b 3 2 6 6 6 6 6 2 6 3 6 5 Rewrite = 4 ⋅ a ⋅ b as a Get a 6 3 5 radical common = 16a b denominator - this is going

Example: Simplify each expression 1 3 1 1 3 1 + + x ⋅x ⋅x = x 2 4 5 2 4 5 10 15 4 29Remember + + =x we add 20 20 20 =x 20 exponents 20 9 20 9 = x ⋅x20 20 =x x

Example: Simplify each expression 4 4 1 −4 1 1 1 w 5 5 5 5 = = 4 = 4 ⋅ 1 w w 5 w5 w 5 w 1 1 1 5 w 5 w 5 w w 5 rationalize To = 4 1 = = = the + 5 w w denominator w5 5 w 5 we want an

Example: Simplify each expression 1 7 −1 1 8 1 x y 8 8 = x = x⋅ 1 = 1 ⋅ 7 xy y y 8 y y 8 8 7 7 xy 8x y 8 To rationalize = = the y y denominator we want an

Example: Simplify each expression 1 1 (2 ) = 2 5 10 5 10 32 32 10 10 = 1 = 1 2 (2 ) 2 8 4 2 8 1 4 8 8 1 1 2 1 1 2 2 − − 4 = 1 =2 2 4 =2 4 4 =2 = 2 4 2 4

Example: Simplify each expression −1 1 1 − − 5 2 5 1 5 2 2 = 1 = ⋅ 1 2 5 2 2 2⋅ 5 2 5 1 1 1 − − 1 −1 1 1 1 = ⋅5 2 2 = ⋅5 = ⋅ = 2 2 2 5 10

Example: Simplify each expression 1 Multiply by 1 m + 1 conjugate and 2 1 ⋅ 1 use FOIL 2 m − 1 m + 1 2 1 m +1 2 m +1 = = m− 1 m− 1

Example: Simplify each expression −2 −2 −2 3 −2 2x 3 −2 2x = 2x 2 2 −3 = 2 x2 x −2 −1 −1 ⋅−2 1 x = 2x = 2 x = 2 ⋅ x = 2 −2 2 2 4

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