How to Factor Algebraic Expressions Containing Fractional and Negative Exponents?

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A polynomial is made of terms in which the exponents, if any, are positive integers. In contrast, more advanced expressions can have fractional and/or negative exponents. For fractional exponents, the numerator acts like a regular exponent, and the denominator dictates the type of root. Negative exponents act like regular exponents except that they move the term across the fraction bar, the line separating the numerator from the denominator. Factoring expressions with fractional or negative exponents requires you to know how to manipulate fractions in addition to knowing how to factor expressions.

    Circle any terms with negative exponents. Rewrite those terms with positive exponents and move the term to the other side the fraction bar. For example, x^-3 becomes 1/(x^3) and 2/(x^-3) becomes 2(x^3). So, to factor 6(xz)^(2/3) - 4/[x^(-3/4)], the first step is to rewrite it as 6(xz)^(2/3) - 4x^(3/4).

    Identify the largest common factor of all of the coefficients. For example, in 6(xz)^(2/3) - 4x^(3/4), 2 is the common factor of the coefficients (6 and 4).

    Divide each term by the common factor from Step 2. Write the quotient next to the factor and separate them with brackets. For example, factoring out a 2 from 6(xz)^(2/3) - 4x^(3/4) yields the following: 2[3(xz)^(2/3) - 2x^(3/4)].

    Identify any variables that appear in every term of the quotient. Circle the term in which that variable is raised to the smallest exponent. In 2[3(xz)^(2/3) - 2x^(3/4)], x appears in every term of the quotient, while z does not. You would circle 3(xz)^(2/3) because 2/3 is less than 3/4.

    Factor out the variable raised to the small power found in Step 4, but not its coefficient. When dividing exponents, find the difference of the two powers and use that as the exponent in the quotient. Use a common denominator when finding the difference of two fractions. In the example above, x^(3/4) divided by x^(2/3) = x^(3/4 - 2/3) = x^(9/12 - 8/12) = x ^(1/12).

    Write the result from Step 5 next to the other factors. Use brackets or parentheses to separate each factor. For example, factoring 6(xz)^(2/3) - 4/[x^(-3/4)] ultimately yields (2)[x^(2/3)][3z^(2/3) - 2x^(1/12)].


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Sly Tutor has been a writer since 2005 and has had work appear in the "Altoona Mirror" newspaper. She holds a Bachelor of Science in microbiology from Pennsylvania State University.

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