Precalculus Examples

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

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

Step 1

Multiply the numerator and denominator of the fraction by

x y

$x y$ .

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Step 1.1

Multiply

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

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

\frac{x y}{x y}

$\frac{x y}{x y}$ .

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

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

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

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

Step 2

Apply the distributive property.

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

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of

x

$x$ .

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Step 3.1.1

Factor

x

$x$ out of

x y

$x y$ .

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

$y$ .

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