Multiplying Rational Fractions with Two Variables: A Step‑by‑Step Guide

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By Amy Harris – Updated Aug 30, 2022

A rational fraction is any fraction whose denominator is non‑zero. In algebra, these fractions contain variables—letters that stand for unknown values. They can be simple monomials (one term in the numerator and denominator) or more complex polynomials with multiple terms. Most learners find multiplication of algebraic fractions easier than addition or subtraction.

Monomials

Multiply the numeric coefficients. Treat the numbers attached to variables as coefficients and the stand‑alone numbers as constants. For example, in (4x²)/(5y) × (3)/(8xy³), multiply 4 × 3 = 12 for the numerator and 5 × 8 = 40 for the denominator.
Combine like variables. Multiply variables with the same base by adding their exponents. Here, the numerator has only x²; the denominator combines y × y³ = y⁴, giving xy⁴.
Form the product. Place the results together: (12x²)/(40xy⁴).
Reduce coefficients. Simplify the numeric fraction by dividing by the greatest common divisor. The example reduces to (3x²)/(10xy⁴).
Cancel variable exponents. Subtract the smaller exponent from the larger for each variable. For x: 2 − 1 = 1, leaving x in the numerator. The final simplified form is (3x)/(10y⁴).

Polynomials

Factor each numerator and denominator. For (x² + x − 2)/(x² + 2x) × (y − 3)/(x² − 2x + 1), factor to [(x − 1)(x + 2)]/[x(x + 2)] × (y − 3)/[(x − 1)(x − 1)].
Cancel common factors. Cross‑cancel identical factors across numerators and denominators: (x + 2) cancels in the first fraction; one (x − 1) cancels between the first numerator and the second denominator. The expression becomes 1/x × (y − 3)/(x − 1).
Multiply the remaining terms. Multiply numerators together and denominators together to get (y − 3)/[x(x − 1)].
Expand if necessary. Remove parentheses: the result is (y − 3)/(x² − x), with the restriction that x ≠ 0 and x ≠ 1.