Solve 9xy/x+9y*(x/9y-9y/x)

$$\frac{ 9xy }{ x+9y } \times ( \frac{ x }{ 9y } - \frac{ 9y }{ x } )$$

Evaluate

$$x-9y$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9y and x is 9xy. Multiply \frac{x}{9y} times \frac{x}{x}. Multiply \frac{9y}{x} times \frac{9y}{9y}.

$$\frac{9xy}{x+9y}\left(\frac{xx}{9xy}-\frac{9y\times 9y}{9xy}\right)$$

Since \frac{xx}{9xy} and \frac{9y\times 9y}{9xy} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9xy}{x+9y}\times \frac{xx-9y\times 9y}{9xy}$$

Do the multiplications in xx-9y\times 9y.

$$\frac{9xy}{x+9y}\times \frac{x^{2}-81y^{2}}{9xy}$$

Multiply \frac{9xy}{x+9y} times \frac{x^{2}-81y^{2}}{9xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9xy\left(x^{2}-81y^{2}\right)}{\left(x+9y\right)\times 9xy}$$

Cancel out 9xy in both numerator and denominator.

$$\frac{x^{2}-81y^{2}}{x+9y}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-9y\right)\left(x+9y\right)}{x+9y}$$

Cancel out x+9y in both numerator and denominator.

$$x-9y$$

Expand

$$x-9y$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9y and x is 9xy. Multiply \frac{x}{9y} times \frac{x}{x}. Multiply \frac{9y}{x} times \frac{9y}{9y}.

$$\frac{9xy}{x+9y}\left(\frac{xx}{9xy}-\frac{9y\times 9y}{9xy}\right)$$

Since \frac{xx}{9xy} and \frac{9y\times 9y}{9xy} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9xy}{x+9y}\times \frac{xx-9y\times 9y}{9xy}$$

Do the multiplications in xx-9y\times 9y.

$$\frac{9xy}{x+9y}\times \frac{x^{2}-81y^{2}}{9xy}$$

Multiply \frac{9xy}{x+9y} times \frac{x^{2}-81y^{2}}{9xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9xy\left(x^{2}-81y^{2}\right)}{\left(x+9y\right)\times 9xy}$$

Cancel out 9xy in both numerator and denominator.

$$\frac{x^{2}-81y^{2}}{x+9y}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-9y\right)\left(x+9y\right)}{x+9y}$$

Cancel out x+9y in both numerator and denominator.

$$x-9y$$