Simplify ((x^2-16y^2)/(xy))/(1/y-4/x)

$\frac{\frac{x^{2} - 16 y^{2}}{x y}}{\frac{1}{y} - \frac{4}{x}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 16 y^{2}}{x y} \cdot \frac{1}{\frac{1}{y} - \frac{4}{x}}$

Step 2

Simplify the numerator.

Rewrite $16 y^{2}$ as ${(4 y)}^{2}$ .

$\frac{x^{2} - {(4 y)}^{2}}{x y} \cdot \frac{1}{\frac{1}{y} - \frac{4}{x}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4 y$ .

$\frac{(x + 4 y) (x - (4 y))}{x y} \cdot \frac{1}{\frac{1}{y} - \frac{4}{x}}$

Multiply $4$ by $- 1$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{1}{y} - \frac{4}{x}}$

Step 3

Simplify the denominator.

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{1}{y} \cdot \frac{x}{x} - \frac{4}{x}}$

To write $- \frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{1}{y} \cdot \frac{x}{x} - \frac{4}{x} \cdot \frac{y}{y}}$

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

Multiply $\frac{1}{y}$ by $\frac{x}{x}$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{x}{y x} - \frac{4}{x} \cdot \frac{y}{y}}$

Multiply $\frac{4}{x}$ by $\frac{y}{y}$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{x}{y x} - \frac{4 y}{x y}}$

Reorder the factors of $y x$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{x}{x y} - \frac{4 y}{x y}}$

Combine the numerators over the common denominator.

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{1}{\frac{x - 4 y}{x y}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{(x + 4 y) (x - 4 y)}{x y} (1 \frac{x y}{x - 4 y})$

Step 5

Multiply $\frac{x y}{x - 4 y}$ by $1$ .

$\frac{(x + 4 y) (x - 4 y)}{x y} \cdot \frac{x y}{x - 4 y}$

Step 6

Cancel the common factor of $x - 4 y$ .

Factor $x - 4 y$ out of $(x + 4 y) (x - 4 y)$ .

$\frac{(x - 4 y) (x + 4 y)}{x y} \cdot \frac{x y}{x - 4 y}$

Cancel the common factor.

$\frac{(x - 4 y) (x + 4 y)}{x y} \cdot \frac{x y}{x - 4 y}$

Rewrite the expression.

$\frac{x + 4 y}{x y} (x y)$

Step 7

Cancel the common factor of $x y$ .

Cancel the common factor.

$\frac{x + 4 y}{x y} (x y)$

Rewrite the expression.

$x + 4 y$

Simplify ((x^2-16y^2)/(xy))/(1/y-4/x)

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