Write each expression as one fraction
Whole numbers go over 1 and mixed-looking expressions stay factored: 2x/3 and 6/x² are already single fractions; a whole like 3 becomes 3/1.
Algebraic fractions
Find the Product of Rational Algebraic Expressions
“Find the product” means multiply — and fraction multiplication never needs common denominators. Multiply across, cancel the shared factors (early is easiest), and note which values the original denominators excluded.
- of means multiply
- multiply across
- cancel before you multiply
- simplify the product
Quick answer
2x/3 × 6/x² = 4/x
Multiply across: (2x · 6)/(3 · x²) = 12x/(3x²). The shared factor 3x cancels, leaving 4/x — with x ≠ 0, because the original denominator x² forbids it.
Cancelling before multiplying reaches the same place with smaller numbers: 6 ÷ 3 = 2 and one x cancels, leaving (2x · 2)/x² = 4/x. Either order, the same product.
The method
Four steps: write, multiply, cancel, exclude
Multiply numerators and denominators across
The product of 2x/3 and 6/x² starts as (2x · 6)/(3 · x²) = 12x/(3x²). Multiplication never needs common denominators.
Cancel common factors — early if you can
Cancel before multiplying to keep numbers small: 12x/(3x²) shares the factor 3x, leaving 4/x. Factor polynomials (like x² = x · x) so the shared factors show.
State the excluded values and finish
Any value that zeroes an original denominator is excluded: in 2x/3 × 6/x², x ≠ 0. The simplified product 4/x is then the finished answer.
Worked example
A coefficient and a power: 2x/3 × 6/x²
2x/3 × 6/x² = 12x/(3x²) = 4/x, x ≠ 0
Multiply across — 12x over 3x² — then cancel the shared factor 3x. Equivalently, cancel first: 6 ÷ 3 = 2 and one x cancels, leaving (2x · 2)/x² = 4/x. Both orders land on the same simplified product.
Worked example
A binomial cancels completely: (x + 2)/5 × 10/(x + 2)
(x + 2)/5 × 10/(x + 2) = 10/5 = 2, x ≠ −2
The binomial (x + 2) sits in both a numerator and a denominator, so it cancels across the products; the 10/5 also cancels to 2. The answer is the plain number 2 — with x = −2 still excluded by the original bar.
Worked example
Powers that mostly cancel: x²/4 × 8/x
x²/4 × 8/x = 8x²/(4x) = 2x, x ≠ 0
Multiply across to 8x²/4x, then cancel 4 and one x. Fast route: 8 ÷ 4 = 2 and x² ÷ x = x. The product of two fractions smaller-looking than the answer is no problem — the sizes of the factors decide.
Worked example
Everything cancels: 3x/4 × 2/(9x)
3x/4 × 2/(9x) = 6x/(36x) = 1/6, x ≠ 0
The full product 6x/36x shares the factor 6x — the entire numerator — so it reduces to 1/6. When the numerator cancels completely, the product is a plain numeric fraction no matter what x was.
Why “of” and “product” point at the easiest operation
Multiplication is the one fraction operation with no common-denominator step. The question “what is half of x/3?” is the product 1/2 × x/3 = x/6 — the word “of” simply names the multiplication. That makes products the friendliest rational-expression task: write each expression as a single fraction, multiply across, and let cancellation do the tidying.
The most satisfying products cancel completely. x/5 × 5/x = 1 for every x ≠ 0, because each variable appears once on top and once on the bottom — the fractions are reciprocals. Recognising reciprocal pairs (and binomial pairs like (x + 2) over its own copy) turns a page of algebra into a one-line answer.
Review reciprocals — the pairs that multiply to 1Common mistakes when finding products
- Hunting for common denominators: Multiplication never needs matching denominators — that is addition and subtraction territory. 2x/3 × 6/x² multiplies straight across; finding the LCD of 3 and x² wastes the operation’s simplicity.
- Cancelling terms instead of factors: In (x + 2)/5 × 10/(x + 2), the (x + 2) cancels because it is a factor of a product. In a sum like x + 2, no cancellation exists until the expression is factored — crossing out matching pieces of sums changes the value.
- Multiplying out before simplifying: Expanding 12x/(3x²) into a bigger polynomial buries the shared factor 3x. Keep products factored, cancel, and expand only if the question demands a particular form.
Practice routine
Cancel early, cancel completely
For each product, factor everything, cancel every shared factor you can see, and only then multiply what remains. Note the excluded values the original denominators carried.
TryWhat to noticeAnswer
x/5 × 5/xEverything cancels — the product is 1.1
4x/9 × 3/(2x)Cancel the x and the 3 before multiplying.2/3
x²/6 × 3/xx² ÷ x leaves one x.x/2
(x − 1)/4 × 8/(x − 1)The binomial is a factor of both sides.2
2/(x²) × x/6One x cancels; 2/6 simplifies to 1/3.1/(3x)
Keep learning
Related fraction topics
Common questions
Finding the Product FAQ
Multiply the numerators together and the denominators together — no common denominators — then cancel the common factors and simplify. For example, 2x/3 × 6/x² = 12x/(3x²) = 4/x, with x ≠ 0. Factoring first makes the cancellations visible.