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

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

Multiply the numerator by the reciprocal of the denominator.

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

Simplify the denominator.

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

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

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

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

Write each expression with a common denominator of $y x$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{x \cdot x + y \cdot y}{x y}$ in a factored form.

Multiply $x$ by $x$ .

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

Multiply $y$ by $y$ .

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

Simplify terms.

Combine.

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

Multiply $y$ by $1$ .

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

Combine $x$ and $\frac{x^{2} + y^{2}}{x y}$ .

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

Reduce the expression $\frac{x (x^{2} + y^{2})}{x y}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

Combine $y$ and $\frac{y}{x^{2} + y^{2}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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