Pre-Algebra Examples

$\frac{10}{a - b} - \frac{6 b}{a^{2} - b^{2}}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$\frac{10}{a - b} - \frac{6 b}{(a + b) (a - b)}$

Step 2

To write $\frac{10}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{10}{a - b} \cdot \frac{a + b}{a + b} - \frac{6 b}{(a + b) (a - b)}$

Step 3

Write each expression with a common denominator of $(a - b) (a + b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{10}{a - b}$ by $\frac{a + b}{a + b}$ .

$\frac{10 (a + b)}{(a - b) (a + b)} - \frac{6 b}{(a + b) (a - b)}$

Step 3.2

Reorder the factors of $(a - b) (a + b)$ .

$\frac{10 (a + b)}{(a + b) (a - b)} - \frac{6 b}{(a + b) (a - b)}$

Step 4

Combine the numerators over the common denominator.

$\frac{10 (a + b) - 6 b}{(a + b) (a - b)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $2$ out of $10 (a + b) - 6 b$ .

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Step 5.1.1

Factor $2$ out of $10 (a + b)$ .

$\frac{2 (5 (a + b)) - 6 b}{(a + b) (a - b)}$

Step 5.1.2

Factor $2$ out of $- 6 b$ .

$\frac{2 (5 (a + b)) + 2 (- 3 b)}{(a + b) (a - b)}$

Step 5.1.3

Factor $2$ out of $2 (5 (a + b)) + 2 (- 3 b)$ .

$\frac{2 (5 (a + b) - 3 b)}{(a + b) (a - b)}$

Step 5.2

Apply the distributive property.

$\frac{2 (5 a + 5 b - 3 b)}{(a + b) (a - b)}$

Step 5.3

Subtract $3 b$ from $5 b$ .

$\frac{2 (5 a + 2 b)}{(a + b) (a - b)}$