Solve (1/xy+y^{2}-6/x^2+xy+9y/x^3+x^2y):(x/y^2-frac{6}{y}+frac{9}{x})

$$( \frac { 1 } { x y + y ^ { 2 } } - \frac { 6 } { x ^ { 2 } + x y } + \frac { 9 y } { x ^ { 3 } + x ^ { 2 } y } ) : ( \frac { x } { y ^ { 2 } } - \frac { 6 } { y } + \frac { 9 } { x } )$$

Evaluate

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

Short Solution Steps

Factor xy+y^{2}. Factor x^{2}+xy.

$$\frac{\frac{1}{y\left(x+y\right)}-\frac{6}{x\left(x+y\right)}+\frac{9y}{x^{3}+x^{2}y}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y\left(x+y\right) and x\left(x+y\right) is xy\left(x+y\right). Multiply \frac{1}{y\left(x+y\right)} times \frac{x}{x}. Multiply \frac{6}{x\left(x+y\right)} times \frac{y}{y}.

$$\frac{\frac{x}{xy\left(x+y\right)}-\frac{6y}{xy\left(x+y\right)}+\frac{9y}{x^{3}+x^{2}y}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Since \frac{x}{xy\left(x+y\right)} and \frac{6y}{xy\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-6y}{xy\left(x+y\right)}+\frac{9y}{x^{3}+x^{2}y}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Factor x^{3}+x^{2}y.

$$\frac{\frac{x-6y}{xy\left(x+y\right)}+\frac{9y}{\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of xy\left(x+y\right) and \left(x+y\right)x^{2} is y\left(x+y\right)x^{2}. Multiply \frac{x-6y}{xy\left(x+y\right)} times \frac{x}{x}. Multiply \frac{9y}{\left(x+y\right)x^{2}} times \frac{y}{y}.

$$\frac{\frac{\left(x-6y\right)x}{y\left(x+y\right)x^{2}}+\frac{9yy}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Since \frac{\left(x-6y\right)x}{y\left(x+y\right)x^{2}} and \frac{9yy}{y\left(x+y\right)x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-6y\right)x+9yy}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Do the multiplications in \left(x-6y\right)x+9yy.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and y is y^{2}. Multiply \frac{6}{y} times \frac{y}{y}.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6y}{y^{2}}+\frac{9}{x}}$$

Since \frac{x}{y^{2}} and \frac{6y}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x-6y}{y^{2}}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and x is xy^{2}. Multiply \frac{x-6y}{y^{2}} times \frac{x}{x}. Multiply \frac{9}{x} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{\left(x-6y\right)x}{xy^{2}}+\frac{9y^{2}}{xy^{2}}}$$

Since \frac{\left(x-6y\right)x}{xy^{2}} and \frac{9y^{2}}{xy^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{\left(x-6y\right)x+9y^{2}}{xy^{2}}}$$

Do the multiplications in \left(x-6y\right)x+9y^{2}.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x^{2}-6yx+9y^{2}}{xy^{2}}}$$

Divide \frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}} by \frac{x^{2}-6yx+9y^{2}}{xy^{2}} by multiplying \frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}} by the reciprocal of \frac{x^{2}-6yx+9y^{2}}{xy^{2}}.

$$\frac{\left(x^{2}-6yx+9y^{2}\right)xy^{2}}{y\left(x+y\right)x^{2}\left(x^{2}-6yx+9y^{2}\right)}$$

Cancel out xy\left(x^{2}-6xy+9y^{2}\right) in both numerator and denominator.

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

Use the distributive property to multiply x by x+y.

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

Expand

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

Short Solution Steps

Factor xy+y^{2}. Factor x^{2}+xy.

$$\frac{\frac{1}{y\left(x+y\right)}-\frac{6}{x\left(x+y\right)}+\frac{9y}{x^{3}+x^{2}y}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y\left(x+y\right) and x\left(x+y\right) is xy\left(x+y\right). Multiply \frac{1}{y\left(x+y\right)} times \frac{x}{x}. Multiply \frac{6}{x\left(x+y\right)} times \frac{y}{y}.

$$\frac{\frac{x}{xy\left(x+y\right)}-\frac{6y}{xy\left(x+y\right)}+\frac{9y}{x^{3}+x^{2}y}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Since \frac{x}{xy\left(x+y\right)} and \frac{6y}{xy\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x-6y}{xy\left(x+y\right)}+\frac{9y}{x^{3}+x^{2}y}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Factor x^{3}+x^{2}y.

$$\frac{\frac{x-6y}{xy\left(x+y\right)}+\frac{9y}{\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of xy\left(x+y\right) and \left(x+y\right)x^{2} is y\left(x+y\right)x^{2}. Multiply \frac{x-6y}{xy\left(x+y\right)} times \frac{x}{x}. Multiply \frac{9y}{\left(x+y\right)x^{2}} times \frac{y}{y}.

$$\frac{\frac{\left(x-6y\right)x}{y\left(x+y\right)x^{2}}+\frac{9yy}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Since \frac{\left(x-6y\right)x}{y\left(x+y\right)x^{2}} and \frac{9yy}{y\left(x+y\right)x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-6y\right)x+9yy}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

Do the multiplications in \left(x-6y\right)x+9yy.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6}{y}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and y is y^{2}. Multiply \frac{6}{y} times \frac{y}{y}.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x}{y^{2}}-\frac{6y}{y^{2}}+\frac{9}{x}}$$

Since \frac{x}{y^{2}} and \frac{6y}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x-6y}{y^{2}}+\frac{9}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and x is xy^{2}. Multiply \frac{x-6y}{y^{2}} times \frac{x}{x}. Multiply \frac{9}{x} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{\left(x-6y\right)x}{xy^{2}}+\frac{9y^{2}}{xy^{2}}}$$

Since \frac{\left(x-6y\right)x}{xy^{2}} and \frac{9y^{2}}{xy^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{\left(x-6y\right)x+9y^{2}}{xy^{2}}}$$

Do the multiplications in \left(x-6y\right)x+9y^{2}.

$$\frac{\frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}}}{\frac{x^{2}-6yx+9y^{2}}{xy^{2}}}$$

Divide \frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}} by \frac{x^{2}-6yx+9y^{2}}{xy^{2}} by multiplying \frac{x^{2}-6yx+9y^{2}}{y\left(x+y\right)x^{2}} by the reciprocal of \frac{x^{2}-6yx+9y^{2}}{xy^{2}}.

$$\frac{\left(x^{2}-6yx+9y^{2}\right)xy^{2}}{y\left(x+y\right)x^{2}\left(x^{2}-6yx+9y^{2}\right)}$$

Cancel out xy\left(x^{2}-6xy+9y^{2}\right) in both numerator and denominator.

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

Use the distributive property to multiply x by x+y.

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