Rational Exponents

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Rational Exponents

  1. 1. Rational Exponents
  2. 2. 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
  3. 3. NOTE: There are 3 different ways to write a rational exponent ( ) 4 4 3 4 3 27 = 27 = 3 27
  4. 4. 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
  5. 5. Evaluating and Simplifying Expressions with Rational Exponents
  6. 6. Evaluate vs. Simplify Evaluate – finding the numerical value. Simplify – writing the expression in simplest form.
  7. 7. 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
  8. 8. In this case, we use the laws of exponents to simplify expressions with rational exponents.
  9. 9. 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
  10. 10. 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
  11. 11. 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
  12. 12. Exampl e: Write as a Radical and Evaluate 1 49 = 49 = 2 2 49 = 49 = 7
  13. 13. 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
  14. 14. 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
  15. 15. 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
  16. 16. 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
  17. 17. 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
  18. 18. 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
  19. 19. 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
  20. 20. 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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