Three moves, one habit. Top stays amber, bottom stays blue. Division is just multiplication wearing a disguise — flip the second fraction and the ÷ becomes a ·.
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.
9.3.1 Multiplying — straight across
To multiply two fractions you multiply the tops together and the bottoms together. Nothing has to match first — no common denominator, no fuss:
The rule AB · CD = ACBD. Just as “23of a half” means 23·12, multiplying takes a part of a part.
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:
Worked example
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).
Watch the holes
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.
🎮 Try itMULTIPLY & CROSS-CANCEL
Click matching factors across the diagonal to cancel them (they turn green and strike through). When you’ve cancelled all you can, see how small the product becomes.
Product:
9.3.2 Dividing — keep, change, flip
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:
AB ÷ CD = AB · DC. The C that was on top is now a bottom — and a bottom can’t be 0.
A new restriction appears
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.
Worked example
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.
🎮 Try itKEEP–CHANGE–FLIP
Press Flip the 2nd to watch the second fraction turn over and the ÷ become a ·. The restriction tracker shows the no-go list growing as the flip drags a numerator down into a denominator.
Problem:
9.3.3 Reduce while you multiply or divide
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
Worked example — the full routine
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.)
The two classic slips
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.
🎮 Try itEFFORT METER: CANCEL FIRST vs MULTIPLY FIRST
Slide between the two strategies for the same product. See how “multiply first” balloons the polynomials while “cancel first” keeps them tiny — same answer, far less work.
Strategy:
9.3.4 Raising to a power — hit top and bottom
A power on a fraction lands on both floors. Since AB·AB = A·AB·B, repeating it n times gives:
(AB)n = AnBn. The exponent copies onto the top and the bottom — never just one.
Three quick ones
• (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.
Mind the parentheses (callback to 5.6)
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.
🎮 Try itPOWER DISTRIBUTOR
Pick a base and an exponent. Watch the exponent copy itself onto the top and the bottom, and see how an even power tames a negative sign.
Base:
Exponent n:
2
9.3.5 Mixing powers, ×, and ÷
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:
Worked example — power, then × and ÷
(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).
🎮 Try itMIXED-ORDER STEPPER
Walk one mixed expression through the order of operations one beat at a time: power → flip → cancel → result. Watch the no-go list grow when the ÷ flips a numerator downstairs.
★ The big ideas, in one breath
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.
What’s next
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.
✎ Exercises 9.3
Factor first, cancel, and state the excluded values. Use a real minus sign in your answers.
Multiply: 3x−1 · x−16
Show answer
The (x−1) cancels and 3/6 trims to 1/2: 12. Restriction: x ≠ 1.
Multiply: x²x+3 · x+3x
Show answer
Cancel (x+3) and one x: x²x = x. Restrictions: x ≠ 0, x ≠ −3.
Multiply: x²−9x+2 · x+2x−3
Show answer
Factor x²−9 = (x−3)(x+3). Cancel (x+2) and (x−3): x+3. Restrictions: x ≠ −2, x ≠ 3.
Divide: x+4x² ÷ x+4x
Show answer
Flip the second to xx+4: (x+4)x·x · x(x+4) = 1x. Restrictions: x ≠ 0, x ≠ −4.
Divide: x²−4x ÷ x+2x
Show answer
Flip: (x−2)(x+2)x · x(x+2) = x−2. Restrictions: x ≠ 0, x ≠ −2.
Full routine: x²−1x²+4x+4 · x+2x−1
Show answer
Factor: (x−1)(x+1)(x+2)² · x+2x−1. Cancel (x−1) and one (x+2): x+1x+2. Restrictions: x ≠ −2, x ≠ 1.
Power: (5x)2
Show answer
Exponent onto both floors: 5²x² = 25x². Restriction: x ≠ 0.
Power: (−2x)3
Show answer
(−2)³ = −8, so (−2)³x³ = −8x³. An odd power keeps the sign negative. Restriction: x ≠ 0.
Power: (x−13)2
Show answer
(x−1)²3² = (x−1)²9, or expanded x²−2x+19. (No variable in the bottom, so no excluded value.)
Mixed order: (2x)2 · x8
Show answer
Power first: 4x². Then multiply: 4·xx·x·8 = 48x = 12x. Restriction: x ≠ 0.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.