Algebra Examples

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Factor $y$ out of $x y x^{2} + y$ .

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Step 3.1.1

Factor $y$ out of $x y x^{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Factor $y$ out of $y^{1}$ .

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

Step 3.1.4

Factor $y$ out of $y (x \cdot x^{2}) + y \cdot 1$ .

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

Step 3.2

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

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Add $1$ and $2$ .

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

Step 3.3

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

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

Step 3.4

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 $a = x$ and $b = 1$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Multiply $- 1$ by $1$ .

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

Step 3.5.2

One to any power is one.

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

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