Rational expressions worksheet

Every question here cancels exactly, because the shared factor is chosen before the question is written. Factor the top and the bottom, cancel what matches, and stop.

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Sigma Prep · sigmaprep.io/worksheets Name Rational Expressions Date Period Simplify each expression. Simplify rational expressions $\frac{a^{2} + 5 a - 84}{a^{2} - 15 a + 56}$ $\frac{k^{2} - 7 k - 60}{k^{2} - 21 k + 108}$ $\frac{a^{2} - 4 a - 21}{a^{2} - 3 a - 28}$ $\frac{x^{2} + 15 x + 56}{x^{2} + 6 x - 7}$ Multiply rational expressions $\frac{p^{2} + 4 p - 12}{p - 3} \cdot \frac{p - 6}{p - 2}$ $\frac{y^{2} + 11 y - 12}{y + 4} \cdot \frac{y + 7}{y - 1}$ $\frac{y^{2} - 1}{y + 4} \cdot \frac{y + 11}{y - 1}$ $\frac{n^{2} + 16 n + 48}{n - 2} \cdot \frac{n - 9}{n + 12}$ Divide rational expressions $\frac{x^{2} - 13 x + 30}{x - 6} \div \frac{x - 3}{x + 10}$ $\frac{y^{2} + 14 y + 24}{y - 12} \div \frac{y + 12}{y - 5}$ $\frac{x^{2} - x - 110}{x - 2} \div \frac{x + 10}{x - 5}$ $\frac{x^{2} + 9 x + 20}{x - 12} \div \frac{x + 4}{x + 1}$ Simplify a quotient of monomials $\frac{- 12 x^{5} y}{12 x^{4} y^{3}}$ $\frac{- 4 x^{4} y^{3}}{8 y^{5}}$ $\frac{- 24 x^{6} y^{5}}{8 x^{3} y^{4}}$ $\frac{- 18 x^{4} y^{2}}{24 x^{4} y}$ Multiply and divide monomial fractions $\frac{9 x^{3}}{3 x y} \cdot \frac{- 2 x^{2}}{7 x}$ $\frac{3 x y^{3}}{2 x} \cdot \frac{5}{3 y}$ $\frac{3 x^{2} y}{9 x^{2} y} \cdot \frac{- 5 x^{3}}{7 y^{2}}$ $\frac{- 9 x^{2}}{6 y^{2}} \div \frac{- 2 x^{2}}{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational Expressions

Date Period

Simplify each expression.

Simplify rational expressions
$\frac{a^{2} + 5 a - 84}{a^{2} - 15 a + 56}$
$\frac{k^{2} - 7 k - 60}{k^{2} - 21 k + 108}$
$\frac{a^{2} - 4 a - 21}{a^{2} - 3 a - 28}$
$\frac{x^{2} + 15 x + 56}{x^{2} + 6 x - 7}$

Multiply rational expressions
$\frac{p^{2} + 4 p - 12}{p - 3} \cdot \frac{p - 6}{p - 2}$
$\frac{y^{2} + 11 y - 12}{y + 4} \cdot \frac{y + 7}{y - 1}$
$\frac{y^{2} - 1}{y + 4} \cdot \frac{y + 11}{y - 1}$
$\frac{n^{2} + 16 n + 48}{n - 2} \cdot \frac{n - 9}{n + 12}$

Divide rational expressions
$\frac{x^{2} - 13 x + 30}{x - 6} \div \frac{x - 3}{x + 10}$
$\frac{y^{2} + 14 y + 24}{y - 12} \div \frac{y + 12}{y - 5}$
$\frac{x^{2} - x - 110}{x - 2} \div \frac{x + 10}{x - 5}$
$\frac{x^{2} + 9 x + 20}{x - 12} \div \frac{x + 4}{x + 1}$

Simplify a quotient of monomials
$\frac{- 12 x^{5} y}{12 x^{4} y^{3}}$
$\frac{- 4 x^{4} y^{3}}{8 y^{5}}$
$\frac{- 24 x^{6} y^{5}}{8 x^{3} y^{4}}$
$\frac{- 18 x^{4} y^{2}}{24 x^{4} y}$

Multiply and divide monomial fractions
$\frac{9 x^{3}}{3 x y} \cdot \frac{- 2 x^{2}}{7 x}$
$\frac{3 x y^{3}}{2 x} \cdot \frac{5}{3 y}$
$\frac{3 x^{2} y}{9 x^{2} y} \cdot \frac{- 5 x^{3}}{7 y^{2}}$
$\frac{- 9 x^{2}}{6 y^{2}} \div \frac{- 2 x^{2}}{6}$



Sigma Prep · sigmaprep.io/worksheets Name Rational ExpressionsAnswer keyVersion 1 Date Period Simplify each expression. Simplify rational expressions $\frac{a^{2} + 5 a - 84}{a^{2} - 15 a + 56}$ $\frac{a + 12}{a - 8}$ $\frac{k^{2} - 7 k - 60}{k^{2} - 21 k + 108}$ $\frac{k + 5}{k - 9}$ $\frac{a^{2} - 4 a - 21}{a^{2} - 3 a - 28}$ $\frac{a + 3}{a + 4}$ $\frac{x^{2} + 15 x + 56}{x^{2} + 6 x - 7}$ $\frac{x + 8}{x - 1}$ Multiply rational expressions $\frac{p^{2} + 4 p - 12}{p - 3} \cdot \frac{p - 6}{p - 2}$ $\frac{(p + 6) (p - 6)}{p - 3}$ $\frac{y^{2} + 11 y - 12}{y + 4} \cdot \frac{y + 7}{y - 1}$ $\frac{(y + 12) (y + 7)}{y + 4}$ $\frac{y^{2} - 1}{y + 4} \cdot \frac{y + 11}{y - 1}$ $\frac{(y + 1) (y + 11)}{y + 4}$ $\frac{n^{2} + 16 n + 48}{n - 2} \cdot \frac{n - 9}{n + 12}$ $\frac{(n + 4) (n - 9)}{n - 2}$ Divide rational expressions $\frac{x^{2} - 13 x + 30}{x - 6} \div \frac{x - 3}{x + 10}$ $\frac{(x - 10) (x + 10)}{x - 6}$ $\frac{y^{2} + 14 y + 24}{y - 12} \div \frac{y + 12}{y - 5}$ $\frac{(y + 2) (y - 5)}{y - 12}$ $\frac{x^{2} - x - 110}{x - 2} \div \frac{x + 10}{x - 5}$ $\frac{(x - 11) (x - 5)}{x - 2}$ $\frac{x^{2} + 9 x + 20}{x - 12} \div \frac{x + 4}{x + 1}$ $\frac{(x + 5) (x + 1)}{x - 12}$ Simplify a quotient of monomials $\frac{- 12 x^{5} y}{12 x^{4} y^{3}}$ $- \frac{x}{y^{2}}$ $\frac{- 4 x^{4} y^{3}}{8 y^{5}}$ $- \frac{x^{4}}{2 y^{2}}$ $\frac{- 24 x^{6} y^{5}}{8 x^{3} y^{4}}$ $- 3 x^{3} y$ $\frac{- 18 x^{4} y^{2}}{24 x^{4} y}$ $- \frac{3 y}{4}$ Multiply and divide monomial fractions $\frac{9 x^{3}}{3 x y} \cdot \frac{- 2 x^{2}}{7 x}$ $- \frac{6 x^{3}}{7 y}$ $\frac{3 x y^{3}}{2 x} \cdot \frac{5}{3 y}$ $\frac{5 y^{2}}{2}$ $\frac{3 x^{2} y}{9 x^{2} y} \cdot \frac{- 5 x^{3}}{7 y^{2}}$ $- \frac{5 x^{3}}{21 y^{2}}$ $\frac{- 9 x^{2}}{6 y^{2}} \div \frac{- 2 x^{2}}{6}$ $\frac{9}{2 y^{2}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational ExpressionsAnswer keyVersion 1

Date Period

Simplify each expression.

Simplify rational expressions
$\frac{a^{2} + 5 a - 84}{a^{2} - 15 a + 56}$ $\frac{a + 12}{a - 8}$
$\frac{k^{2} - 7 k - 60}{k^{2} - 21 k + 108}$ $\frac{k + 5}{k - 9}$
$\frac{a^{2} - 4 a - 21}{a^{2} - 3 a - 28}$ $\frac{a + 3}{a + 4}$
$\frac{x^{2} + 15 x + 56}{x^{2} + 6 x - 7}$ $\frac{x + 8}{x - 1}$

Multiply rational expressions
$\frac{p^{2} + 4 p - 12}{p - 3} \cdot \frac{p - 6}{p - 2}$ $\frac{(p + 6) (p - 6)}{p - 3}$
$\frac{y^{2} + 11 y - 12}{y + 4} \cdot \frac{y + 7}{y - 1}$ $\frac{(y + 12) (y + 7)}{y + 4}$
$\frac{y^{2} - 1}{y + 4} \cdot \frac{y + 11}{y - 1}$ $\frac{(y + 1) (y + 11)}{y + 4}$
$\frac{n^{2} + 16 n + 48}{n - 2} \cdot \frac{n - 9}{n + 12}$ $\frac{(n + 4) (n - 9)}{n - 2}$

Divide rational expressions
$\frac{x^{2} - 13 x + 30}{x - 6} \div \frac{x - 3}{x + 10}$ $\frac{(x - 10) (x + 10)}{x - 6}$
$\frac{y^{2} + 14 y + 24}{y - 12} \div \frac{y + 12}{y - 5}$ $\frac{(y + 2) (y - 5)}{y - 12}$
$\frac{x^{2} - x - 110}{x - 2} \div \frac{x + 10}{x - 5}$ $\frac{(x - 11) (x - 5)}{x - 2}$
$\frac{x^{2} + 9 x + 20}{x - 12} \div \frac{x + 4}{x + 1}$ $\frac{(x + 5) (x + 1)}{x - 12}$

Simplify a quotient of monomials
$\frac{- 12 x^{5} y}{12 x^{4} y^{3}}$ $- \frac{x}{y^{2}}$
$\frac{- 4 x^{4} y^{3}}{8 y^{5}}$ $- \frac{x^{4}}{2 y^{2}}$
$\frac{- 24 x^{6} y^{5}}{8 x^{3} y^{4}}$ $- 3 x^{3} y$
$\frac{- 18 x^{4} y^{2}}{24 x^{4} y}$ $- \frac{3 y}{4}$

Multiply and divide monomial fractions
$\frac{9 x^{3}}{3 x y} \cdot \frac{- 2 x^{2}}{7 x}$ $- \frac{6 x^{3}}{7 y}$
$\frac{3 x y^{3}}{2 x} \cdot \frac{5}{3 y}$ $\frac{5 y^{2}}{2}$
$\frac{3 x^{2} y}{9 x^{2} y} \cdot \frac{- 5 x^{3}}{7 y^{2}}$ $- \frac{5 x^{3}}{21 y^{2}}$
$\frac{- 9 x^{2}}{6 y^{2}} \div \frac{- 2 x^{2}}{6}$ $\frac{9}{2 y^{2}}$

How to do these

$\frac{x^{2} + 5 x + 6}{x^{2} + 2 x - 3}$

Factor the top: (x + 2)(x + 3).
Factor the bottom: (x + 3)(x - 1).
The (x + 3) cancels.

The answer is $\frac{x + 2}{x - 1}$ .

Where students go wrong

Canceling terms instead of factors. In (x + 6)/(x + 2) the x does not cancel, because it is added, not multiplied. Nothing can be canceled until both the top and the bottom are fully factored.

Also called algebraic fractions.

What you can put on this worksheet

Simplify by canceling a common factor: $\frac{v^{2} + 2 v - 24}{v^{2} + 8 v + 12}$ → $\frac{v - 4}{v + 2}$
Multiplying two rational expressions: $\frac{v^{2} + 2 v - 24}{v + 2} \cdot \frac{v + 3}{v - 4}$ → $\frac{(v + 6) (v + 3)}{v + 2}$
Dividing, by flipping the second one: $\frac{v^{2} + 2 v - 24}{v + 2} \div \frac{v - 4}{v + 3}$ → $\frac{(v + 6) (v + 3)}{v + 2}$
Simplify and state the excluded values: Simplify, and state the excluded values:
$\frac{v^{2} - 2 v - 35}{v^{2} - 10 v + 21}$ → $\frac{v + 5}{v - 3}$ ; $v \neq 3, 7$
Monomials: $\frac{6 x^{4}}{12 x^{5}}$ → $\frac{1}{2 x}$
Monomials only: $\frac{6 y}{7 x^{2}} \cdot \frac{4 x^{2} y^{3}}{8 y^{2}}$ → $\frac{3 y^{2}}{7}$

Questions about these worksheets

Yes. Every rational expressions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 1 is usually taken around 8th or 9th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the rational expressions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: simplify by canceling a common factor, multiplying two rational expressions, dividing, by flipping the second one, simplify and state the excluded values, monomials and monomials only. Tick as many as you want and set how many of each, or let it spread them evenly.

Canceling terms instead of factors. In (x + 6)/(x + 2) the x does not cancel, because it is added, not multiplied. Nothing can be canceled until both the top and the bottom are fully factored.

Rational expressions worksheet

Select your difficulty

Simplify rational expressions

Multiply rational expressions

Divide rational expressions

Simplify a quotient of monomials

Multiply and divide monomial fractions

Simplify rational expressions

Multiply rational expressions

Divide rational expressions

Simplify a quotient of monomials

Multiply and divide monomial fractions

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these rational expressions worksheets free?+

Do the rational expressions worksheets come with an answer key?+

Are these rational expressions worksheets printable?+

What grade are these rational expressions worksheets for?+

Can I get a different version of the same rational expressions worksheet?+

Is this a free alternative to Kuta Software for rational expressions?+

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What do students get wrong with rational expressions?+