Precalculus Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x y$ .

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Step 1.1

Multiply $\frac{x + \frac{y}{x}}{y + \frac{x}{y}}$ by $\frac{x y}{x y}$ .

$\frac{x y}{x y} \cdot \frac{x + \frac{y}{x}}{y + \frac{x}{y}}$

Step 1.2

Combine.

$\frac{x y (x + \frac{y}{x})}{x y (y + \frac{x}{y})}$

Step 2

Apply the distributive property.

$\frac{x y x + x y \frac{y}{x}}{x y \cdot y + x y \frac{x}{y}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

Cancel the common factor.

$\frac{x y x + x y \frac{y}{x}}{x y \cdot y + x y \frac{x}{y}}$

Step 3.1.3

Rewrite the expression.

$\frac{x y x + y \cdot y}{x y \cdot y + x y \frac{x}{y}}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $y$ .

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Step 3.6.1

Factor $y$ out of $x y$ .

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

Step 3.6.2

Cancel the common factor.

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

Step 3.6.3

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Factor $y$ out of $x y x$ .

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

Step 4.1.2

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

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

Step 4.1.3

Factor $y$ out of $y (x \cdot x) + y \cdot y$ .

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

Step 4.2

Multiply $x$ by $x$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

Factor $x$ out of $x y \cdot y$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x$ out of $x (y \cdot y) + x \cdot x$ .

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

Step 5.2

Multiply $y$ by $y$ .

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