Lesson 7 · from §66 of the 1910 text
Multiplying Fractions
Just as in arithmetic, the product of fractions is the product of the numerators over the product of the denominators: \frac34 \cdot \frac57 = \frac{15}{28}, \qquad \frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd}, \qquad n\cdot\frac{a}{b} = \frac{na}{b}.
The smart move is to cancel before you multiply.
Example 1. \dfrac{4a^2x^3}{5y^3}\cdot\dfrac{35y^5}{16a^3x^2}. Cancel 4, 5, a^2, x^2, y^3: the result is \dfrac{7xy^2}{4a}.
Example 2. Factor everything first: \frac{x^2 - 4}{x^2 - 5x + 6}\cdot\frac{x + 2}{x + 3}\cdot\frac{2x^3 - 18x}{x^2 + 4x + 4} = \frac{(x + 2)(x - 2)}{(x - 3)(x - 2)}\cdot\frac{x + 2}{x + 3}\cdot\frac{2x(x + 3)(x - 3)}{(x + 2)^2} = 2x. Check with x = 1: \frac{-3}{2}\cdot\frac34\cdot\frac{-16}{9} = 2. ✓
Example 3. With a mixed expression, turn it into a fraction first: 3a - \dfrac{b^2 - 3ab^2}{3a - b^2} = \dfrac{9a^2 - b^2}{3a - b^2}. Watch for factors like a - b and b - a, which differ only by a factor of -1.
Rule. Change whole and mixed expressions to fractions. Factor every numerator and denominator, cancel factors common to any numerator and any denominator, and multiply what's left.
Quick check
\displaystyle \frac{x^2-1}{x^2+2x}\cdot\frac{x}{x-1}
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Simplify each product. Cancel common factors before you multiply.
\displaystyle \frac{3x^2y^3}{4a^4}\cdot\frac{8a^3}{12xy}
\displaystyle \frac{10a^3b^2}{15c^3}\cdot\frac{18c}{7a^2}
\displaystyle \frac{3x}{8y}\cdot\frac{6ax}{9b}\cdot\frac{12b^2y}{16x}
\displaystyle \frac{14m^2n^4}{5n}\cdot\frac{25n^3}{6m^2}\cdot\frac{4x}{7m}
\displaystyle \left(\frac{2x}{3ay}\right)^2\cdot\frac{12a^3}{16b^2xy^3}\cdot 8b^4y^2
\displaystyle \frac{a}{4c^2}\cdot\left(\frac{c}{2a}\right)^3\cdot 16
\displaystyle \left(\frac{2a}{c}\right)^2\cdot\left(\frac{2c}{a}\right)^3\cdot\left(\frac{1}{4}\right)^2
\displaystyle \left(\frac{-3a}{3}\right)^2\cdot\frac{c^5}{(2a)^5}\cdot\left(\frac{2}{3}\right)^2
\displaystyle \frac{6c}{a}\cdot\left(\frac{9c^2}{4a}\right)^2\cdot\left(\frac{-2a}{3c}\right)^3
\displaystyle \frac{(4ax)^2}{225c^4}\cdot\frac{(5ac^2)^3}{(2x^2)^3}\cdot\left(\frac{-x}{c}\right)^5
\displaystyle \frac{2a+4x}{5y^2}\cdot\frac{15y}{a+2x}
\displaystyle \frac{3c}{9-c^2}\cdot\frac{c^2+5c+6}{18cd}
\displaystyle \frac{3a-6y}{4a+2y}\cdot\frac{8(2a+y)^2}{4a^2-24ay+24y^2}
\displaystyle \frac{c^2+6ce+9e^2}{6(c+d)^2}\cdot\frac{3c^2e-12d^2e}{c^2-2cd+3ce-6de}
\displaystyle \frac{a^2+4ab+4b^2}{9-a^2}\cdot\frac{a^2-5a+6}{a^2-4b^2}
\displaystyle \frac{5c^2-20d^2}{c^3+8d^3}\cdot\frac{c^2-2cd+4d^2}{25cd^4}
Mixed and whole expressions: turn them into fractions first, as in Example 3.
\displaystyle \frac{(x+2)^2}{3x^3+6x^2+12x}\cdot(15-9x^2)\cdot\frac{x^3-8}{4-x^2}
\displaystyle \frac{8c-24d}{c^2-6cd+9d^2}\cdot\left(4c-\frac{d^2}{c}\right)\cdot\frac{4c^2+d^2}{64c^4-4d^4}
\displaystyle \left(\frac{6m-9}{2m-3}+2m\right)\left(2m-9+\frac{36}{2m+3}\right)
\displaystyle \left(\frac{3}{2a}-1\right)^2\cdot\frac{8a^2x}{9-4a^2}\cdot\frac{3+2a}{2a-3}
\displaystyle \left(1+\frac{2}{x}-\frac{3}{x^2}\right)\cdot\frac{9x^3}{3(x^2-18x+17)}
\displaystyle \frac{x^3-8}{6xy}\cdot\left(3-\frac{4}{2-x}\right)\cdot\frac{18x^2y^2}{x^2+2x+4}
\displaystyle \frac{a^3-b^3}{(a-b)^3}\cdot\frac{6a+6b}{2a^2+2ab+2b^2}\cdot\frac{(a-b)^2}{a^2-b^2}
\displaystyle \left(4-\frac{20-9x}{5-x}-x\right)\left(\frac{1}{x}+\frac{1}{x^2}-\frac{30}{x^3}\right)
\displaystyle \left(x+2y-\frac{5x+10y}{x+y}\right)\left(\frac{y^2+x^2+2yx}{x^2-3xy-4y^2}\right)\cdot\frac{1}{x+2y}