1Cademy - Adding $$\frac{5}{12x^2y} + \frac{4}{21xy^2}$$

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How to Add or Subtract Rational Expressions

Example

Adding $\frac{5}{12x^2y} + \frac{4}{21xy^2}$

Add two rational expressions whose denominators are monomials containing multiple variables:

$\frac{5}{12x^2y} + \frac{4}{21xy^2}$

Step 1 — Find the LCD of $12x^2y$ and $21xy^2$ . Factor each denominator into primes and variable factors: $12x^2y = 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y$ and $21xy^2 = 3 \cdot 7 \cdot x \cdot y \cdot y$ . Take the highest power of every factor that appears: $\text{LCD} = 2 \cdot 2 \cdot 3 \cdot 7 \cdot x^2 \cdot y^2 = 84x^2y^2$ .

Step 2 — Rewrite each fraction with the LCD as its denominator. Compare each denominator to the LCD to identify missing factors. The denominator $12x^2y$ is missing $7y$ , so multiply by $\frac{7y}{7y}$ . The denominator $21xy^2$ is missing $4x$ , so multiply by $\frac{4x}{4x}$ :

$\frac{5 \cdot 7y}{12x^2y \cdot 7y} + \frac{4 \cdot 4x}{21xy^2 \cdot 4x} = \frac{35y}{84x^2y^2} + \frac{16x}{84x^2y^2}$

Step 3 — Add the numerators over the common denominator:

$\frac{16x + 35y}{84x^2y^2}$

Step 4 — Check for simplification. The numerator $16x + 35y$ shares no common factor with the denominator $84x^2y^2$ , so the fraction is already in simplified form.

This example extends the addition procedure to rational expressions with monomial denominators. Finding the LCD uses the same prime-and-variable factoring technique as for numerical fractions, but each variable base is also treated as a factor — the LCD includes each variable raised to the highest exponent that appears in any denominator.

Updated 2026-04-21

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OpenStax Elementary Algebra

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