What Is a Method for Multiplying Fractions With Exponents?

# What Is a Method for Multiplying Fractions With Exponents?

A good method for multiplying fractions with exponents is to factor, cancel and simplify. This works whether the exponent is attached to a variable or a numeral.

One example in which there are no variables is "(3^2 / 4^2) x (2^4 / 6^3)." In this case, the first step of factoring simply involves writing out all the exponents: "(3 x 3 / 4 x 4) x (2 x 2 x 2 x 2 / 6 x 6 x 6)," and then factoring again: "(3 x 3 / 2 x 2 x 2 x 2) x (2 x 2 x 2 x 2 / 2 x 3 x 2 x 3 x 2 x 3)." The second step requires canceling. All the numerators cancel out with two of the 3s and four of the 2s in the denominator, leaving only "1 / 2 x 2 x 2 x 3." The third step is to simplify, so in this case the denominator gets multiplied for the final answer: "1 / 24."

An example of a problem with variables is "[(x^2 + 3x + 2) / (2x^2 + 7x +3)] x [(2x^2 + 9x + 9) / (x^2 + 3x + 2)]." First, it needs to be factored: "[(x + 2) (x + 1) / (x + 3) (2x + 1)] x [(2x + 3) (x + 3) / (x + 2) (x + 1)]." Canceling comes next. Because there is an "(x + 2)," an "(x + 1)" and an "(x + 3)" in both the numerator and the denominator, all of them cancel each other out, leaving only "(2x + 3) / (2x + 1)." In this case, there is no simplifying to be done, so "(2x + 3) / (2x + 1)" is the final answer.

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