Solve (3/x^2+x/2)div(1/6-1/x)

$$( \frac { 3 } { x ^ { 2 } } + \frac { x } { 2 } ) \div ( \frac { 1 } { 6 } - \frac { 1 } { x } )$$

Evaluate

$$\frac{3\left(x^{3}+6\right)}{x\left(x-6\right)}$$

Short Solution Steps

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

$$\frac{\frac{3\times 2}{2x^{2}}+\frac{xx^{2}}{2x^{2}}}{\frac{1}{6}-\frac{1}{x}}$$

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

$$\frac{\frac{3\times 2+xx^{2}}{2x^{2}}}{\frac{1}{6}-\frac{1}{x}}$$

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

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

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

$$\frac{\frac{6+x^{3}}{2x^{2}}}{\frac{x}{6x}-\frac{6}{6x}}$$

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

$$\frac{\frac{6+x^{3}}{2x^{2}}}{\frac{x-6}{6x}}$$

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

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

Cancel out 2x in both numerator and denominator.

$$\frac{3\left(x^{3}+6\right)}{x\left(x-6\right)}$$

Use the distributive property to multiply 3 by x^{3}+6.

$$\frac{3x^{3}+18}{x\left(x-6\right)}$$

Use the distributive property to multiply x by x-6.

$$\frac{3x^{3}+18}{x^{2}-6x}$$

Expand

$$\frac{3\left(x^{3}+6\right)}{x\left(x-6\right)}$$

Short Solution Steps

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

$$\frac{\frac{3\times 2}{2x^{2}}+\frac{xx^{2}}{2x^{2}}}{\frac{1}{6}-\frac{1}{x}}$$

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

$$\frac{\frac{3\times 2+xx^{2}}{2x^{2}}}{\frac{1}{6}-\frac{1}{x}}$$

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

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

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

$$\frac{\frac{6+x^{3}}{2x^{2}}}{\frac{x}{6x}-\frac{6}{6x}}$$

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

$$\frac{\frac{6+x^{3}}{2x^{2}}}{\frac{x-6}{6x}}$$

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

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

Cancel out 2x in both numerator and denominator.

$$\frac{3\left(x^{3}+6\right)}{x\left(x-6\right)}$$

Use the distributive property to multiply 3 by x^{3}+6.

$$\frac{3x^{3}+18}{x\left(x-6\right)}$$

Use the distributive property to multiply x by x-6.

$$\frac{3x^{3}+18}{x^{2}-6x}$$