Operations with Algebraic Fractions

The main operations involving algebraic fractions are addition and subtraction, multiplication and division, and exponentiation.

Addition

When two algebraic fractions share the same denominator, their sum is another algebraic fraction whose numerator is the sum of the numerators, while the denominator stays the same.

$$ \frac{A}{B} + \frac{C}{B} = \frac{A+C}{B} $$

Example. $$ \frac{4x-3}{2y} + \frac{2x}{2y} = \frac{4x-3+2x}{2y} = \frac{6x-3}{2y} $$

Subtraction

Similarly, if two algebraic fractions have the same denominator, their difference is an algebraic fraction with a numerator equal to the difference of the numerators, and the same denominator as before.

$$ \frac{A}{B} - \frac{C}{B} = \frac{A-C}{B} $$

Example. $$ \frac{4x-3}{2y} - \frac{2x}{2y} = \frac{4x-3-2x}{2y} = \frac{2x-3}{2y} $$

Multiplication

The product of two algebraic fractions is obtained by multiplying the numerators together to form the new numerator, and multiplying the denominators together to form the new denominator.

$$ \frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D} $$

Example. $$ \frac{4x-3}{2y} \cdot \frac{2x}{3y} = \frac{(4x-3) \cdot 2x}{2y \cdot 3y} = \frac{(4x-3) \cdot x}{3y^2} = \frac{4x^2-3x}{3y^2} $$

Division

To divide one algebraic fraction by another, multiply the first fraction by the reciprocal of the second.

$$ \frac{A}{B} \ : \ \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} $$

Example. $$ \frac{4x-3}{2y} \ : \ \frac{2x}{3y} = \frac{4x-3}{2y} \cdot \frac{3y}{2x} = \frac{(4x-3) \cdot 3y}{2y \cdot 2x} = \frac{(4x-3) \cdot 3}{4x} = \frac{12x-9}{4x} $$

Exponentiation

Raising an algebraic fraction to the nth power means raising both its numerator and denominator to the nth power.

$$ \left(\frac{A}{B}\right)^2 = \frac{A^2}{B^2} $$

Example. $$ \left(\frac{4x-3}{2y}\right)^2 = \frac{(4x-3)^2}{(2y)^2} $$

And the same reasoning extends to higher powers.