Solve x/2y^2+y/3x^2/frac{1{6xy}+2/x^2y}

$$\frac { \frac { x } { 2 y ^ { 2 } } + \frac { y } { 3 x ^ { 2 } } } { \frac { 1 } { 6 x y } + \frac { 2 } { x ^ { 2 } y } }$$

Evaluate

$$\frac{3x^{3}+2y^{3}}{y\left(x+12\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2y^{2} and 3x^{2} is 6x^{2}y^{2}. Multiply \frac{x}{2y^{2}} times \frac{3x^{2}}{3x^{2}}. Multiply \frac{y}{3x^{2}} times \frac{2y^{2}}{2y^{2}}.

$$\frac{\frac{x\times 3x^{2}}{6x^{2}y^{2}}+\frac{y\times 2y^{2}}{6x^{2}y^{2}}}{\frac{1}{6xy}+\frac{2}{x^{2}y}}$$

Since \frac{x\times 3x^{2}}{6x^{2}y^{2}} and \frac{y\times 2y^{2}}{6x^{2}y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x\times 3x^{2}+y\times 2y^{2}}{6x^{2}y^{2}}}{\frac{1}{6xy}+\frac{2}{x^{2}y}}$$

Do the multiplications in x\times 3x^{2}+y\times 2y^{2}.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{1}{6xy}+\frac{2}{x^{2}y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6xy and x^{2}y is 6yx^{2}. Multiply \frac{1}{6xy} times \frac{x}{x}. Multiply \frac{2}{x^{2}y} times \frac{6}{6}.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{x}{6yx^{2}}+\frac{2\times 6}{6yx^{2}}}$$

Since \frac{x}{6yx^{2}} and \frac{2\times 6}{6yx^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{x+2\times 6}{6yx^{2}}}$$

Do the multiplications in x+2\times 6.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{x+12}{6yx^{2}}}$$

Divide \frac{3x^{3}+2y^{3}}{6x^{2}y^{2}} by \frac{x+12}{6yx^{2}} by multiplying \frac{3x^{3}+2y^{3}}{6x^{2}y^{2}} by the reciprocal of \frac{x+12}{6yx^{2}}.

$$\frac{\left(3x^{3}+2y^{3}\right)\times 6yx^{2}}{6x^{2}y^{2}\left(x+12\right)}$$

Cancel out 6yx^{2} in both numerator and denominator.

$$\frac{3x^{3}+2y^{3}}{y\left(x+12\right)}$$

Use the distributive property to multiply y by x+12.

$$\frac{3x^{3}+2y^{3}}{yx+12y}$$

Expand

$$\frac{3x^{3}+2y^{3}}{y\left(x+12\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2y^{2} and 3x^{2} is 6x^{2}y^{2}. Multiply \frac{x}{2y^{2}} times \frac{3x^{2}}{3x^{2}}. Multiply \frac{y}{3x^{2}} times \frac{2y^{2}}{2y^{2}}.

$$\frac{\frac{x\times 3x^{2}}{6x^{2}y^{2}}+\frac{y\times 2y^{2}}{6x^{2}y^{2}}}{\frac{1}{6xy}+\frac{2}{x^{2}y}}$$

Since \frac{x\times 3x^{2}}{6x^{2}y^{2}} and \frac{y\times 2y^{2}}{6x^{2}y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x\times 3x^{2}+y\times 2y^{2}}{6x^{2}y^{2}}}{\frac{1}{6xy}+\frac{2}{x^{2}y}}$$

Do the multiplications in x\times 3x^{2}+y\times 2y^{2}.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{1}{6xy}+\frac{2}{x^{2}y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6xy and x^{2}y is 6yx^{2}. Multiply \frac{1}{6xy} times \frac{x}{x}. Multiply \frac{2}{x^{2}y} times \frac{6}{6}.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{x}{6yx^{2}}+\frac{2\times 6}{6yx^{2}}}$$

Since \frac{x}{6yx^{2}} and \frac{2\times 6}{6yx^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{x+2\times 6}{6yx^{2}}}$$

Do the multiplications in x+2\times 6.

$$\frac{\frac{3x^{3}+2y^{3}}{6x^{2}y^{2}}}{\frac{x+12}{6yx^{2}}}$$

Divide \frac{3x^{3}+2y^{3}}{6x^{2}y^{2}} by \frac{x+12}{6yx^{2}} by multiplying \frac{3x^{3}+2y^{3}}{6x^{2}y^{2}} by the reciprocal of \frac{x+12}{6yx^{2}}.

$$\frac{\left(3x^{3}+2y^{3}\right)\times 6yx^{2}}{6x^{2}y^{2}\left(x+12\right)}$$

Cancel out 6yx^{2} in both numerator and denominator.

$$\frac{3x^{3}+2y^{3}}{y\left(x+12\right)}$$

Use the distributive property to multiply y by x+12.

$$\frac{3x^{3}+2y^{3}}{yx+12y}$$