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Study Guides > College Algebra

Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, 2x14+5x342{x}^{\frac{1}{4}}+5{x}^{\frac{3}{4}} can be factored by pulling out x14{x}^{\frac{1}{4}} and being rewritten as x14(2+5x12){x}^{\frac{1}{4}}\left(2+5{x}^{\frac{1}{2}}\right).

Example 7: Factoring an Expression with Fractional or Negative Exponents

Factor 3x(x+2)−13+4(x+2)233x{\left(x+2\right)}^{\frac{-1}{3}}+4{\left(x+2\right)}^{\frac{2}{3}}.

Solution

Factor out the term with the lowest value of the exponent. In this case, that would be (x+2)−13{\left(x+2\right)}^{-\frac{1}{3}}.
(x+2)−13(3x+4(x+2))Factor out the GCF.(x+2)−13(3x+4x+8)Simplify.(x+2)−13(7x+8)\begin{array}{cc}{\left(x+2\right)}^{-\frac{1}{3}}\left(3x+4\left(x+2\right)\right)\hfill & \text{Factor out the GCF}.\hfill \\ {\left(x+2\right)}^{-\frac{1}{3}}\left(3x+4x+8\right)\hfill & \text{Simplify}.\hfill \\ {\left(x+2\right)}^{-\frac{1}{3}}\left(7x+8\right)\hfill & \end{array}

Try It 8

Factor 2(5a−1)34+7a(5a−1)−142{\left(5a - 1\right)}^{\frac{3}{4}}+7a{\left(5a - 1\right)}^{-\frac{1}{4}}. Solution

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