The lowest common denominator (LCD) is the smallest expression that both denominators can divide into evenly. This is often found by factoring each denominator, then combining all unique factors with their highest exponents.

Practice Problems

1. State the lowest common denominator:

$$\frac{3}{8}, \quad \frac{57}{72}$$

2. State the lowest common denominator:

$$\frac{-2}{3x^3 y^4}, \quad \frac{5}{9x^2 y^6}$$

3. State the lowest common denominator:

$$\frac{3}{4x^2 + 12x}, \quad \frac{7}{3x^2 + 9x}$$

4. State the lowest common denominator:

$$\frac{5}{x^2 – 9}, \quad \frac{4}{x^2 – 3x}$$

5. State the lowest common denominator:

$$\frac{2}{5a^2 b^3}, \quad \frac{7}{10a^4 b}$$

6. State the lowest common denominator:

$$\frac{1}{(m+2)^2}, \quad \frac{3}{m^2 – 4}$$

Solutions

1.
The smallest number divisible by both 8 and 72 is 72.

$$\text{LCD} = 72$$

2.
Factor each denominator:

$$3x^3 y^4, \quad 9x^2 y^6$$

Numerical LCD: $9$ (largest factor from $3$ and $9$)
Variable part: $x^3$ (largest power from $x^3$ and $x^2$), $y^6$ (largest power from $y^4$ and $y^6$)

$$\text{LCD} = 9x^3 y^6$$

3.
Factor each:

$$4x^2 + 12x = 4x(x+3)$$ $$3x^2 + 9x = 3x(x+3)$$

Numerical LCD: $12$ (smallest number divisible by 4 and 3)
Variable part: $x(x+3)$

$$\text{LCD} = 12x(x+3)$$

4.
Factor each:

$$x^2 – 9 = (x – 3)(x + 3)$$ $$x^2 – 3x = x(x – 3)$$

LCD must include $x$, $x – 3$, $x + 3$:

$$\text{LCD} = x(x – 3)(x + 3)$$

5.
Numerical LCD: $10$ (LCM of 5 and 10)
Variable part: $a^4$ (largest from $a^2$ and $a^4$), $b^3$ (largest from $b^3$ and $b$)

$$\text{LCD} = 10a^4 b^3$$

6.
Factor each:

$$(m+2)^2$$ $$m^2 – 4 = (m-2)(m+2)$$

LCD must include $(m+2)^2$ (highest power) and $(m-2)$:

$$\text{LCD} = (m+2)^2 (m-2)$$