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How to Simplify Fractions With Variables

Students learn how to simplify fractions with variables during their first-year algebra course, typically taken in eighth or ninth grade. Some prerequisite knowledge is needed to successfully simplify fractions with variables. For instance, students must be able to simplify fractions without variables, a procedure that includes skills such as determining a greatest common factor, or GCF. They also must be familiar with certain terminology, such as that of an exponent, which is a number written in superscript above and to the right of a variable.

Instructions

    • 1

      Reduce the fraction's coefficients to lowest terms. The coefficients are the leading numbers appearing to the left of the variables. To reduce the coefficients to lowest terms, determine the GCF, which is the largest number that multiplies into both, then divide the numerator and denominator by this number, separately. For example, consider the problem [6(a^4)(b^2)c] / [9(a^4)(b^5)]. The coefficients are 6 and 9, and their GCF is 3. Dividing the numerator by 3 yields 2, and dividing the denominator by 3 yields 3, producing [2(a^4)(b^2)c] / [3(a^4)(b^5)].

    • 2

      Cancel any like variables that possess identical exponents. In [2(a^4)(b^2)c] / [3(a^4)(b^5)], the "a" variables have matching exponents of 4. So the "a^4" in the numerator cancels out the "a^4" repeated in the denominator, removing the "a" variables from the expression and thus rendering it [2(b^2)c] / [3(b^5)].

    • 3

      Subtract the exponents of the variables in the denominator from their like variables in the numerator. After performing this subtraction, place variables with positive exponents in the numerator, but place variables with negative exponents in the denominator, changing the negative exponents to a positive ones. In [2(b^2)c] / [3(b^5)], the variable "b" appears in both the numerator and the denominator. Subtract the exponents: 2 - 5 = -3. So you would obtain b^-3. Because this exponent is negative, place it in the denominator, where it becomes positive. Thus, the example simplifies to (2c) / (3b^3). Repeat this process for all variables that are common to both the numerator and the denominator, until there aren't any variables shared by the numerator and denominator. In the example, since no like variables exist among the numerator and denominator, (2c) / (3b^3) is the final answer.

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