Simplify ((x+1/y)^m(x-1/y)^n)/((y+1/x)^m(y-1/x)^n)

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

Simplify the numerator.

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

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

Combine the numerators over the common denominator.

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

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

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

Combine the numerators over the common denominator.

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

Apply the product rule to $\frac{x y + 1}{y}$ .

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

Apply the product rule to $\frac{x y - 1}{y}$ .

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

Simplify the denominator.

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

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

Combine the numerators over the common denominator.

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

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

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

Combine the numerators over the common denominator.

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

Apply the product rule to $\frac{y x + 1}{x}$ .

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

Apply the product rule to $\frac{y x - 1}{x}$ .

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

Combine fractions.

Multiply $\frac{{(x y + 1)}^{m}}{y^{m}}$ by $\frac{{(x y - 1)}^{n}}{y^{n}}$ .

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

Multiply $\frac{{(y x + 1)}^{m}}{x^{m}}$ by $\frac{{(y x - 1)}^{n}}{x^{n}}$ .

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

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Combine.

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

Cancel the common factor of ${(x y + 1)}^{m}$ and ${(y x + 1)}^{m}$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of ${(x y - 1)}^{n}$ and ${(y x - 1)}^{n}$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{m + n}}{y^{m + n}}$

Simplify ((x+1/y)^m(x-1/y)^n)/((y+1/x)^m(y-1/x)^n)

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