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How to Add Fractional Exponents

Fractional exponents are rational values that appear as exponential quantities. They take the form of (n/m) in an exponential of x that appears as x^(n/m). In plain English, this sort of quantity indicates you must "take the nth exponent of x and then take the mth root of the nth exponent of x" or vice versa. Just as x^3 cannot be added to x^2 in variable form, x^(n/m) cannot be added to x^(p/q). However, the product of exponential quantities can be combined using the Law of Exponents.

Things You'll Need

Pencil
Paper

Instructions

Example: Simplify [x^(3/2)][x^(5/7)]
- 1
  Write down the term whose fractional exponents are to be combined. For the current example, the term is written as [x^(3/2)][x^(5/7)].
- 2
  Write down the exponents that appear in the term as a sum of fractions. For our example, this appears as 3/2 + 5/7.
- 3
  Find the least common denominator of the terms that appear in the sum of exponents. The denominators here are 2 and 7. These integers are common factors of 14, which cannot be reduced with respect to both factors any further.
- 4
  Multiply the numerator in each fraction by the factor that produces the same fraction with a denominator of 14. This gives us 3/2(7/7) + 5/7(2/2) = 21/14 + 10/14.
- 5
  Add the numerators atop the common denominator: 21/14 + 10/14 = 31/14.
- 6
  Reduce the rational result as much as possible. Here, 31 and 14 contain no common factors, and therefore the rational remains as already written.
- 7
  Re-write the term as one fully combined entity: [x^(3/2)][x^(5/7)] = [x^(21/14)][x^(10/14)] = x^(31/14).

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