Algebra Examples

$\frac{y}{x^{2}} + \frac{1}{x^{2} y^{2}}$

Step 1

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

$\frac{y}{x^{2}} \cdot \frac{y^{2}}{y^{2}} + \frac{1}{x^{2} y^{2}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{y}{x^{2}}$ by $\frac{y^{2}}{y^{2}}$ .

$\frac{y \cdot y^{2}}{x^{2} y^{2}} + \frac{1}{x^{2} y^{2}}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{y \cdot y^{2} + 1}{x^{2} y^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.1.1

Multiply $y$ by $y^{2}$ .

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Step 3.1.1.1

Raise $y$ to the power of $1$ .

$\frac{y^{1} y^{2} + 1}{x^{2} y^{2}}$

Step 3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{1 + 2} + 1}{x^{2} y^{2}}$

Step 3.1.2

Add $1$ and $2$ .

$\frac{y^{3} + 1}{x^{2} y^{2}}$

Step 3.2

Rewrite $1$ as $1^{3}$ .

$\frac{y^{3} + 1^{3}}{x^{2} y^{2}}$

Step 3.3

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 $a = y$ and $b = 1$ .

$\frac{(y + 1) (y^{2} - y \cdot 1 + 1^{2})}{x^{2} y^{2}}$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $- 1$ by $1$ .

$\frac{(y + 1) (y^{2} - y + 1^{2})}{x^{2} y^{2}}$

Step 3.4.2

One to any power is one.

$\frac{(y + 1) (y^{2} - y + 1)}{x^{2} y^{2}}$