Multiplying & Dividing Rational Expressions

Perform multiplication and division on rational expressions. Grade 11 Math 20-1.

Lesson 6.3 of Rational Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Multiplying Rational Expressions. Factor, state NPVs, write as one fraction, cancel common factors across the multiplication sign, and simplify — applied to rational expressions with monomials and polynomials.
• Dividing Rational Expressions. Apply Keep, Change, Flip to convert division into multiplication, and learn why dividing requires NPVs from three different denominators — including the divisor's numerator.
• Complex Fractions and Mixed Operations. Simplify fractions-within-fractions by collapsing each layer into a single fraction, then dividing — and handle chained multiply-divide expressions by working left to right.
• Working Backwards. Given a target simplified result, determine the missing polynomial in a rational expression by tracing what cancellation must have occurred.

Inside the lesson — a free preview

The Procedure

5 Steps to Multiply Rational Expressions.

Key Insight — Cancelling Across the Multiplication Sign. You can cancel across the multiplication sign. A factor in one numerator can cancel with a factor in the other fraction's denominator. This is a major time-saver — but it only works for multiplication, not for any other operation.

Connection to Numeric Fractions. Numbers: (12)/(10) × (5)/(4) = (12 · 5)/(10 · 4) = (60)/(40) = (3)/(2)

Rational expressions: (x)/(3) · (3x)/(4) = (x · 3x)/(3 · 4) = (3x^2)/(12) = (x^2)/(4)

Same process in both cases — factor, multiply tops and bottoms, cancel, reduce. Variables do not change anything fundamental.

Worked example: a — Monomial Multiplication

Simplify (3c^2)/(10) · (5d^3)/(9c^3).

Example 1a — (3c^2)/(10) · (5d^3)/(9c^3). Step 1 — Find NPVs from the original denominators.

Denominators: 10 (a constant — no NPV) and 9c^3.

9c^3 = 0 → c^3 = 0 → c = 0

So c ≠ 0.

Step 2 — Multiply numerators and denominators together.

(3c^2)/(10) · (5d^3)/(9c^3) = (3c^2 · 5d^3)/(10 · 9c^3) = (15c^2d^3)/(90c^3)

Step 3 — Reduce the numerical coefficients. GCF of 15 and 90 is 15:

(15)/(90) = (1)/(6)

Step 4 — Reduce the variable parts using the exponent rule (subtract exponents):

(c^2)/(c^3) = c^(2-3) = c^(-1) = (1)/(c)

d^3 has no d in the denominator, so it stays.

Step 5 — Combine all parts:

= (d^3)/(6c)

Answer: (d^3)/(6c), where c ≠ 0

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