Practice Multiplying and Dividing Rational Expressions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

Multiplying rational expressions by multiplying numerators together and denominators together (after factoring and canceling). Dividing by multiplying by the reciprocal of the divisor.

It works exactly like multiplying and dividing numeric fractions. To multiply: factor everything, cancel common factors across any numerator and any denominator, then multiply across. To divide: flip the second fraction and multiply. abรทcd=abโ‹…dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}.

Showing a random 20 of 50 problems.

Example 1

easy
Divide 5x2รทx4\frac{5x}{2}\div\frac{x}{4}, xโ‰ 0x\neq 0.

Example 2

easy
State the reciprocal of x+2xโˆ’3\frac{x+2}{x-3} (no restrictions needed).

Example 3

medium
Multiply x2+6x+9x2โˆ’9โ‹…xโˆ’3x+3\frac{x^2 + 6x + 9}{x^2 - 9}\cdot\frac{x-3}{x+3}.

Example 4

easy
Multiply 3xโ‹…x26\frac{3}{x}\cdot\frac{x^2}{6}, xโ‰ 0x\neq0.

Example 5

medium
Divide 2x2+xโˆ’3x2โˆ’1รท2x+3x+1\frac{2x^2 + x - 3}{x^2 - 1}\div\frac{2x + 3}{x + 1}.

Example 6

easy
Multiply x+12โ‹…4x+1\frac{x + 1}{2}\cdot\frac{4}{x + 1}, xโ‰ โˆ’1x\neq-1.

Example 7

medium
Multiply x2โˆ’xโˆ’12x+2โ‹…x+2xโˆ’4\frac{x^2 - x - 12}{x+2}\cdot\frac{x+2}{x-4}.

Example 8

easy
Multiply 2x3โ‹…94x\frac{2x}{3}\cdot\frac{9}{4x}, xโ‰ 0x\neq0.

Example 9

medium
Divide x2โˆ’4x+5รทxโˆ’2x+5\frac{x^2 - 4}{x+5}\div\frac{x-2}{x+5}.

Example 10

easy
Multiply xโˆ’12โ‹…8xโˆ’1\frac{x-1}{2}\cdot\frac{8}{x-1}, xโ‰ 1x\neq 1.

Example 11

hard
Multiply x2โˆ’5x+6x2โˆ’2xโˆ’3โ‹…x2โˆ’9x2โˆ’4\frac{x^2 - 5x + 6}{x^2 - 2x - 3}\cdot\frac{x^2 - 9}{x^2 - 4}.

Example 12

hard
Simplify 3x2+6xx2+5x+6โ‹…x+33x\frac{3x^2 + 6x}{x^2 + 5x + 6}\cdot\frac{x+3}{3x}, state restrictions.

Example 13

easy
Multiply 2xโ‹…x3\frac{2}{x}\cdot\frac{x}{3}, xโ‰ 0x\neq0.

Example 14

easy
What is the restriction for x+4xโˆ’2\frac{x+4}{x-2}?

Example 15

easy
Multiply xx+1โ‹…x+1x2\frac{x}{x+1}\cdot\frac{x+1}{x^2}, xโ‰ 0,โˆ’1x\neq 0,-1.

Example 16

medium
Multiply x2โˆ’1x+3โ‹…x+3x+1\frac{x^2 - 1}{x + 3} \cdot \frac{x + 3}{x + 1}.

Example 17

challenge
Divide x3โˆ’8x2โˆ’4รทx2+2x+4x+2\frac{x^3 - 8}{x^2 - 4}\div\frac{x^2 + 2x + 4}{x + 2}, state restrictions.

Example 18

medium
Multiply x2โˆ’4x+3โ‹…x+3xโˆ’2\frac{x^2 - 4}{x + 3}\cdot\frac{x + 3}{x - 2}, state restrictions.

Example 19

medium
Divide x2+7x+12x+2รทx+3x+2\frac{x^2 + 7x + 12}{x+2}\div\frac{x+3}{x+2}.

Example 20

medium
State all restrictions for the product x+1xโˆ’2โ‹…xโˆ’2x+3\frac{x+1}{x-2}\cdot\frac{x-2}{x+3}.