Multiply straight across, divide by flipping — and factor first so the work stays small.
Point 4 of 5 in this lesson: 9.3.4 Raising to a power — hit top and bottom
You already know how to multiply numeric fractions: 23 of 35 is 615, which trims to 25. Multiply rational expressions and the recipe is identical — top times top, bottom times bottom. The only thing that changes is that the bottoms now carry letters, so they carry their no-go list along with them. Everything from 9.2 pays off here: factor first, cancel, and the answer falls out small.
In this lesson you'll learn to (1) multiply two rational expressions and cancel across them before you ever multiply out, (2) divide by keeping the first, flipping the second, and multiplying — watching the new restriction that flipping creates, (3) run a full factor-cancel-multiply example end to end, and (4) raise a rational expression to a power by sending the exponent to both floors. Throughout, the color code holds: numerator = amber, denominator = blue, a cancelled factor = green (struck through), and an excluded value = red.
To multiply two fractions you multiply the tops together and the bottoms together. Nothing has to match first — no common denominator, no fuss:
The danger isn't the rule — it's the temptation to multiply everything out first. If A, B, C, D are big polynomials, “straight across” hands you a monster you then have to factor and trim anyway. The smart order is the reverse: factor everything, cancel any factor that shows up on a top and a bottom, and only multiply what survives. Watch:
xx+1 · x+1x². The factor (x+1) sits on a top and a bottom, so it cancels. One x on top cancels one of the two on the bottom:
= x · (x+1)(x+1) · x² = 1x
The product is just 1x — and we never expanded a thing. Restrictions: x ≠ 0 and x ≠ −1 (from the original bottoms, even though (x+1) vanished from the answer).
Cancelling a factor makes it disappear from the printed answer, but the value it forbade is still excluded. The original expression has no meaning at x = −1, so the simplified form keeps that ban even though you no longer see (x+1). The factored bottoms tell you the whole no-go list: x ≠ 0, x ≠ −1.
Dividing by a fraction means multiplying by its reciprocal — flip it over. Why? Because dividing by CD asks “how many CD’s fit?”, and that’s the same as taking DC of the thing. Three words: keep, change, flip. Keep the first fraction, change ÷ to ·, flip the second:
When you flip CD into DC, the old numerator C becomes a denominator. So C ≠ 0 joins the no-go list — even though C started life upstairs. Dividing by a fraction quietly forbids the second fraction from being 0. Always read the restrictions off the original problem and the flip.
x²−1x ÷ x+1x². Keep the first, flip the second to x²x+1, and factor x²−1 = (x−1)(x+1):
= (x−1)(x+1)x · x·x(x+1) = x(x−1)
The (x+1) cancels and one x kills the lone x on the floor, leaving x(x−1) = x²−x. Restrictions: x ≠ 0 and x ≠ −1 — the −1 coming from the factor we flipped down into a denominator.
Put the two habits together. Faced with any product or quotient, run the same four-beat routine every time:
| Beat | Do this |
|---|---|
| 1 · Flip | if it’s a ÷, flip the second fraction and make it a · |
| 2 · Factor | factor every top and every bottom completely |
| 3 · Cancel | cancel any factor that appears on a top and a bottom |
| 4 · Multiply | multiply only what survives — that’s your answer |
x²−1x²+4x+4 · x+2x−1
Factor: x²−1 = (x−1)(x+1), and x²+4x+4 = (x+2)². Rewrite:
(x−1)(x+1)(x+2)(x+2) · (x+2)(x−1)
Cancel: the (x−1)’s and one pair of (x+2)’s vanish. What survives:
= x+1x+2
Restrictions: x ≠ −2 and x ≠ 1. (Set each original bottom to 0: x²+4x+4=0 ⟹ x=−2; x−1=0 ⟹ x=1.)
1. Multiplying out before cancelling. Expand (x−1)(x+1)(x+2) over (x+2)²(x−1) and you’re staring at degree-3 polynomials you’ll have to re-factor. Cancel first, always.
2. Cancelling a term, not a factor. You may only cancel whole factors that multiply the entire top and bottom. In x+1x+2 you cannot “cancel the x’s” — those x’s are tied up in sums, not standing alone as factors.
A power on a fraction lands on both floors. Since AB·AB = A·AB·B, repeating it n times gives:
• (2x)3 = 2³x³ = 8x³ (x ≠ 0)
• (x+1x)2 = (x+1)²x² = x²+2x+1x² — leave it factored as (x+1)² unless you’re asked to expand.
• (−2x)2 = (−2)²x² = 4x² — an even power makes a negative base positive.
The exponent only spreads to both floors when the whole fraction is inside parentheses. (−2x)2 = 4x², but −2²x = −4x means something completely different — there the minus sits outside the square and the bottom isn’t squared at all.
When a problem mixes all three, order of operations still rules: do the powers first, then the × and ÷ left to right — flipping for every ÷, and cancelling as you go. Same beats from 9.3.3, just more of them. Here is the full march on one expression:
(2x)2 · x4 ÷ x+1x
1 · Power first: the exponent hits both floors, so (2x)2 = 4x²:
4x² · x4 ÷ x+1x
2 · Flip the ÷: keep the first two, change ÷ to ·, flip the last to xx+1 — and now x ≠ −1 joins the list:
4x² · x4 · xx+1
3 · Cancel left to right: the 4’s cancel, and the two x’s on top kill the x² on the bottom:
= 4 · x · xx·x · 4 · (x+1) = 1x+1
4 · Result: 1x+1. Restrictions: x ≠ 0 and x ≠ −1 (the −1 from the fraction we flipped down).
Multiply straight across — top·top over bottom·bottom — but factor and cancel across the two fractions before you multiply, so you never expand a monster. Divide by keeping the first, changing ÷ to ·, and flipping the second; that flip drags a numerator down into a denominator, so it adds a new restriction (the flipped piece can’t be 0). Raise to a power by sending the exponent to both floors: (AB)n = AnBn. Through it all, cancel only whole factors, never terms across a + or −, and keep every excluded value on the list even after it cancels out of sight.
Multiplying and dividing never needed a common denominator — the bottoms just multiplied. Adding and subtracting do. In 9.4 you’ll match the bottoms first (using the LCD from 9.2), combine the tops, and learn to tame the most dangerous minus sign in all of algebra.
Factor first, cancel, and state the excluded values. Use a real minus sign in your answers.
Six questions to lock it in. Tap the answer you think is right.
This lesson develops CCSS A-APR.D.7 (a (+) standard: multiply and divide rational expressions, treating the system like the rational numbers) and A-SSE.A.2 (use the structure of an expression — here, factoring before operating). It assumes the factoring of Stage 8 and the numeric-fraction intuition of Stage 3.
The #1 misconception is two-headed: students flip the wrong fraction when dividing (or flip both), and they multiply everything out before cancelling, producing high-degree polynomials they then can’t simplify. The antidote: a fixed mantra — “keep the first, flip the second; factor and cancel before you multiply.” Have students say it aloud and write the factored forms before touching pencil to a product. A close companion error is cancelling terms instead of factors (“cancelling the x’s” in (x+1)/(x+2)); insist that only a factor of the whole top and the whole bottom may be struck. Finally, keep restrictions visible: an excluded value that cancels out of the printed answer is still excluded, and division silently adds the restriction that the flipped fraction’s old numerator can’t be 0.