Algebra Examples

$\frac{a^{2} + b^{2}}{a^{3} + b^{3}} - \frac{1}{a + b}$

Step 1

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = a$ and $b = b$ .

$\frac{a^{2} + b^{2}}{(a + b) (a^{2} - a b + b^{2})} - \frac{1}{a + b}$

Step 2

To write $- \frac{1}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a^{2} - a b + b^{2}}{a^{2} - a b + b^{2}}$ .

$\frac{a^{2} + b^{2}}{(a + b) (a^{2} - a b + b^{2})} - \frac{1}{a + b} \cdot \frac{a^{2} - a b + b^{2}}{a^{2} - a b + b^{2}}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{1}{a + b}$ by $\frac{a^{2} - a b + b^{2}}{a^{2} - a b + b^{2}}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- (- a b)$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{a^{2} + b^{2} - a^{2} + 1 (a b) - b^{2}}{(a + b) (a^{2} - a b + b^{2})}$

Step 4.2.2

Multiply $a$ by $1$ .

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

Step 4.3

Subtract $a^{2}$ from $a^{2}$ .

$\frac{b^{2} + 0 + a b - b^{2}}{(a + b) (a^{2} - a b + b^{2})}$

Step 4.4

Add $b^{2}$ and $0$ .

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

Step 4.5

Subtract $b^{2}$ from $b^{2}$ .

$\frac{a b + 0}{(a + b) (a^{2} - a b + b^{2})}$

Step 4.6

Add $a b$ and $0$ .

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