Basic Math Examples

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

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 $b = b$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $a + b$ out of $(a + b) (a^{2} - a b + b^{2}) + (a + b) b^{2}$ .

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Step 4.1.1

Factor $a + b$ out of $(a + b) b^{2}$ .

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

Step 4.1.2

Factor $a + b$ out of $(a + b) (a^{2} - a b + b^{2}) + (a + b) (b^{2})$ .

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

Step 4.2

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

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