Simplify (x+x/y)/(y-y/x)

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $y$ .

Factor $y$ out of $y x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{y}{x}$ into the numerator.

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

Factor $x$ out of $y x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

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

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

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

Multiply by $1$ .

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

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

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

Multiply $x^{- 1}$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x^{- 1}$ .

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

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

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

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

Subtract $1$ from $1$ .

$\frac{x^{2} (y x^{0} + 1)}{y x y - y^{2}}$

Simplify $y x^{0}$ .

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

Step 5

Simplify the denominator.

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

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

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

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

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

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

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

Multiply $y^{- 1}$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y^{- 1}$ .

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

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

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

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

Subtract $1$ from $1$ .

$\frac{x^{2} (y + 1)}{y^{2} (y^{0} x - 1)}$

Simplify $y^{0} x$ .

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

Simplify (x+x/y)/(y-y/x)

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