Fractions as Exponents

• Rational exponent

  a raised to 1/k is equal to the k root of a for any natural number k and any a greater than 0
• Negative rational exponent


  a raised to -m/n is equal to the reciprocal of the n root of a
• Simplifying exponential notations with rational exponents

  1. Copy the base

  2. Add the exponents for multiplication

  3. Subtract the exponents for division

  4. Raise a power to a power by multiplying the exponents
• Simplifying exponential notations

  • 9 raised to 1/2 = square root of 9 = 3
  • 16 raised to 1/4 = fourth root of 16 = 2
  • 27 raised to 1/3 = cube root of 27 = 3
  • 16 raised to 1/4 = fourth root of 16 = 2
• Multiplying rational exponents

  • 36 raised to 1/4 times 36 raised to 1/4 = 36 raised to 1/2 = square root of 36 = 6
  • 16 raised to 1/8 times 16 raised to 1/8 = 16 raised to 1/4 = fourth root of 16 = 2
  • 49 raised to 1/3 times 49 raised to 1/6 = 49 raised to 1/2 = square root of 49 = 7
• Dividing rational exponents

  • 81 raised to 3/4 over 81 raised to 1/2 = 81 raised to 1/4 = fourth root of 81 = 3
  • 8 raised to 2/3 over 8 raised to 1/3 = 8 raised to 1/3 = cube root of 8 = 2
  • 15 raised to 3/4 over 15 raised to 1/2 = 15 raised to 1/2 = square root of 15
• Raising a power to a power

  • 2 raised to (1/3 raised to 3) = 2 raised to 1 = 2
  • 27 raised to (1/3 raised to 2) = 27 raised to 2/3 = cube root of 27 squared = 9
• Simplifying using properties of exponents

  • 729 raised to 1/6 = (9 cubed) raised to 1/6 = 9 raised to 1/2 = square root of 9 = 3
  • 4 times 9 raised to 1/2 = 4 times square root of 9 = 4 times 3 = 12
  • 8 times 64 raised to 1/3 = 8 times cube root of 64 = 8 times 4 = 32
• If m and n are positive integers that are relatively prime, and a is a real number not equal to 0, then a raised to -m/n is equal to the reciprocal of the n root of a