How to Factor With Negative Fractional Exponents

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An expression with negative exponents can appear complicated to factor, but you can simplify the process by rewriting the terms that contain negative fractional exponents. A number with a negative fractional exponent is equivalent to the reciprocal of that number, or 1 over that number, with a positive fractional exponent. With rewritten terms, you can find a greatest common factor, which is the largest term that divides evenly into each term in the expression. An expression that has been simplified and factored is easier to work with and solve than one that contains negative fractional exponents.

  • Determine an expression that contains negative fractional exponents. For example, use the expression x^(-4/3) + 2x^(-1/3).

  • Rewrite each term that contains a negative fractional exponent as a reciprocal with a positive fractional exponent in the denominator. In the example, this results in 1/(x^(4/3)) + 2/(x^(1/3)).

  • Find the greatest common factor of the expression. In the example, the term 1/(x^(1/3)) is the greatest common factor because both terms contain a multiple of x^(1/3) in their denominators.

  • Divide the first term by the greatest common factor, which is equivalent to multiplying by the reciprocal of the greatest common factor. In the example, divide 1/(x^(4/3)) by 1/(x^(1/3)), which is equivalent to 1/(x^(4/3)) times x^(1/3). Cancel out the term x^(1/3) in the numerator and denominator, leaving 1/(x^(3/3)) for the first term.

  • Divide the second term by the greatest common factor, which is equivalent to multiplying by the reciprocal of the greatest common factor. In the example, divide 2/(x^(1/3)) by 1/(x^(1/3)), which is equivalent to 2/(x^(1/3)) times x^(1/3). Cancel out the term x^(1/3) in the numerator and denominator, leaving 2 for the second term.

  • Write the greatest common factor outside brackets that contain the factored first and second terms. In the example, write 1/(x^(1/3))[1/(x^(3/3)) + 2].

  • Simplify or reduce any fractional exponents. In the example, reduce the fractional exponent 3/3 to 1, which eliminates the exponent because a number raised to the power of 1 is the number itself. This leaves 1/(x^(1/3))[1/x + 2], or [1/x + 2]/[x^(1/3)] .

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