Multiplication and Division of Rational Fractions

Product of two ( and in general any finite number) rational fraction equals the fraction with the numerator, that equals the production of denominators of multiplied fractions: `color(red)(P_1/Q_1*P_2/Q_2=(P_1*P_2)/(Q_1*Q_2))` .

The quotient of two rational fractions equals the fraction with the numerator, that equals the product of numerators of first fraction and denominator of second fraction, and the denominator equals product of denominator of first fraction and numerator of second fraction: `color(blue)(P_1/Q_1-:P_2/Q_2=(P_1xxQ_2)/(Q_1xxP_2))` .

In fact: we should factor the numerators and denominators of original fractions before multiplication and division.

Example 1. Perform the following operation: `(x^2+2x+1)/(18x^3)*(9x^4)/(x^2-1)` .

We have `(x^2+2x+1)/(18x^3)=(x+1)^2/(18x^3)` ; `(9x^4)/(x^2-1)=(9x^4)/((x-1)(x+1))` .

Using the rule of fraction multiplication, we obtain:

`(x^2+2x+1)/(18x^3)*(9x^4)/(x^2-1)=((x+1)^2*9x^4)/(18x^3(x+1)(x-1))=(x(x+1))/(2(x-1))` .

Example 2. Perform the following operation: `(a^3-2a^2)/(3a+3)-:(a^2-4)/(3a^2+6a+3)` .

We have `(a^3-2a^2)/(3a+3)=(a^2(a-2))/(3(a+1))`; `(a^2-4)/(3a^2+6a+3)=((a-2)(a+2))/(3(a+1)^2` .

Using the rule of fraction division, we obtain:

`(a^3-2a^2)/(3a+3)-:(a^2-4)/(3a^2+6a+3)=(a^2(a-2)*3(a+1)^2)/(3(a+1)(a-2)(a+2))=(a^2(a+1))/(a+2)` .