Solve (frac{15x^{3}}{y^{4}}-5x/y^2+5/x):(3x^2/y^3-1/y+frac{y}{x^2})

$$( \frac { 15 x ^ { 3 } } { y ^ { 4 } } - \frac { 5 x } { y ^ { 2 } } + \frac { 5 } { x } ) : ( \frac { 3 x ^ { 2 } } { y ^ { 3 } } - \frac { 1 } { y } + \frac { y } { x ^ { 2 } } )$$

Evaluate

$$\frac{5x}{y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{4} and y^{2} is y^{4}. Multiply \frac{5x}{y^{2}} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{15x^{3}}{y^{4}}-\frac{5xy^{2}}{y^{4}}+\frac{5}{x}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

Since \frac{15x^{3}}{y^{4}} and \frac{5xy^{2}}{y^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{15x^{3}-5xy^{2}}{y^{4}}+\frac{5}{x}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{4} and x is xy^{4}. Multiply \frac{15x^{3}-5xy^{2}}{y^{4}} times \frac{x}{x}. Multiply \frac{5}{x} times \frac{y^{4}}{y^{4}}.

$$\frac{\frac{\left(15x^{3}-5xy^{2}\right)x}{xy^{4}}+\frac{5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

Since \frac{\left(15x^{3}-5xy^{2}\right)x}{xy^{4}} and \frac{5y^{4}}{xy^{4}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(15x^{3}-5xy^{2}\right)x+5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

Do the multiplications in \left(15x^{3}-5xy^{2}\right)x+5y^{4}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{3} and y is y^{3}. Multiply \frac{1}{y} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{y^{2}}{y^{3}}+\frac{y}{x^{2}}}$$

Since \frac{3x^{2}}{y^{3}} and \frac{y^{2}}{y^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{2}-y^{2}}{y^{3}}+\frac{y}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{3} and x^{2} is x^{2}y^{3}. Multiply \frac{3x^{2}-y^{2}}{y^{3}} times \frac{x^{2}}{x^{2}}. Multiply \frac{y}{x^{2}} times \frac{y^{3}}{y^{3}}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{\left(3x^{2}-y^{2}\right)x^{2}}{x^{2}y^{3}}+\frac{yy^{3}}{x^{2}y^{3}}}$$

Since \frac{\left(3x^{2}-y^{2}\right)x^{2}}{x^{2}y^{3}} and \frac{yy^{3}}{x^{2}y^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{\left(3x^{2}-y^{2}\right)x^{2}+yy^{3}}{x^{2}y^{3}}}$$

Do the multiplications in \left(3x^{2}-y^{2}\right)x^{2}+yy^{3}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{4}-y^{2}x^{2}+y^{4}}{x^{2}y^{3}}}$$

Divide \frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}} by \frac{3x^{4}-y^{2}x^{2}+y^{4}}{x^{2}y^{3}} by multiplying \frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}} by the reciprocal of \frac{3x^{4}-y^{2}x^{2}+y^{4}}{x^{2}y^{3}}.

$$\frac{\left(15x^{4}-5x^{2}y^{2}+5y^{4}\right)x^{2}y^{3}}{xy^{4}\left(3x^{4}-y^{2}x^{2}+y^{4}\right)}$$

Cancel out xy^{3} in both numerator and denominator.

$$\frac{x\left(15x^{4}-5x^{2}y^{2}+5y^{4}\right)}{y\left(3x^{4}-x^{2}y^{2}+y^{4}\right)}$$

Factor the expressions that are not already factored.

$$\frac{5x\left(3x^{4}-x^{2}y^{2}+y^{4}\right)}{y\left(3x^{4}-x^{2}y^{2}+y^{4}\right)}$$

Cancel out 3x^{4}-x^{2}y^{2}+y^{4} in both numerator and denominator.

$$\frac{5x}{y}$$

Expand

$$\frac{5x}{y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{4} and y^{2} is y^{4}. Multiply \frac{5x}{y^{2}} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{15x^{3}}{y^{4}}-\frac{5xy^{2}}{y^{4}}+\frac{5}{x}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

Since \frac{15x^{3}}{y^{4}} and \frac{5xy^{2}}{y^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{15x^{3}-5xy^{2}}{y^{4}}+\frac{5}{x}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{4} and x is xy^{4}. Multiply \frac{15x^{3}-5xy^{2}}{y^{4}} times \frac{x}{x}. Multiply \frac{5}{x} times \frac{y^{4}}{y^{4}}.

$$\frac{\frac{\left(15x^{3}-5xy^{2}\right)x}{xy^{4}}+\frac{5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

Since \frac{\left(15x^{3}-5xy^{2}\right)x}{xy^{4}} and \frac{5y^{4}}{xy^{4}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(15x^{3}-5xy^{2}\right)x+5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

Do the multiplications in \left(15x^{3}-5xy^{2}\right)x+5y^{4}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{1}{y}+\frac{y}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{3} and y is y^{3}. Multiply \frac{1}{y} times \frac{y^{2}}{y^{2}}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{2}}{y^{3}}-\frac{y^{2}}{y^{3}}+\frac{y}{x^{2}}}$$

Since \frac{3x^{2}}{y^{3}} and \frac{y^{2}}{y^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{2}-y^{2}}{y^{3}}+\frac{y}{x^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{3} and x^{2} is x^{2}y^{3}. Multiply \frac{3x^{2}-y^{2}}{y^{3}} times \frac{x^{2}}{x^{2}}. Multiply \frac{y}{x^{2}} times \frac{y^{3}}{y^{3}}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{\left(3x^{2}-y^{2}\right)x^{2}}{x^{2}y^{3}}+\frac{yy^{3}}{x^{2}y^{3}}}$$

Since \frac{\left(3x^{2}-y^{2}\right)x^{2}}{x^{2}y^{3}} and \frac{yy^{3}}{x^{2}y^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{\left(3x^{2}-y^{2}\right)x^{2}+yy^{3}}{x^{2}y^{3}}}$$

Do the multiplications in \left(3x^{2}-y^{2}\right)x^{2}+yy^{3}.

$$\frac{\frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}}}{\frac{3x^{4}-y^{2}x^{2}+y^{4}}{x^{2}y^{3}}}$$

Divide \frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}} by \frac{3x^{4}-y^{2}x^{2}+y^{4}}{x^{2}y^{3}} by multiplying \frac{15x^{4}-5x^{2}y^{2}+5y^{4}}{xy^{4}} by the reciprocal of \frac{3x^{4}-y^{2}x^{2}+y^{4}}{x^{2}y^{3}}.

$$\frac{\left(15x^{4}-5x^{2}y^{2}+5y^{4}\right)x^{2}y^{3}}{xy^{4}\left(3x^{4}-y^{2}x^{2}+y^{4}\right)}$$

Cancel out xy^{3} in both numerator and denominator.

$$\frac{x\left(15x^{4}-5x^{2}y^{2}+5y^{4}\right)}{y\left(3x^{4}-x^{2}y^{2}+y^{4}\right)}$$

Factor the expressions that are not already factored.

$$\frac{5x\left(3x^{4}-x^{2}y^{2}+y^{4}\right)}{y\left(3x^{4}-x^{2}y^{2}+y^{4}\right)}$$

Cancel out 3x^{4}-x^{2}y^{2}+y^{4} in both numerator and denominator.

$$\frac{5x}{y}$$