Solve c-8/c+3div(c-2/4c+8c^4/c^2+3c)

$$\frac { c - 8 } { c + 3 } \div ( \frac { c - 2 } { 4 c } + \frac { 8 c ^ { 4 } } { c ^ { 2 } + 3 c } )$$

Evaluate

$$\frac{4c\left(c-8\right)}{32c^{4}+c^{2}+c-6}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{8c^{4}}{c^{2}+3c}.

$$\frac{\frac{c-8}{c+3}}{\frac{c-2}{4c}+\frac{8c^{4}}{c\left(c+3\right)}}$$

Cancel out c in both numerator and denominator.

$$\frac{\frac{c-8}{c+3}}{\frac{c-2}{4c}+\frac{8c^{3}}{c+3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4c and c+3 is 4c\left(c+3\right). Multiply \frac{c-2}{4c} times \frac{c+3}{c+3}. Multiply \frac{8c^{3}}{c+3} times \frac{4c}{4c}.

$$\frac{\frac{c-8}{c+3}}{\frac{\left(c-2\right)\left(c+3\right)}{4c\left(c+3\right)}+\frac{8c^{3}\times 4c}{4c\left(c+3\right)}}$$

Since \frac{\left(c-2\right)\left(c+3\right)}{4c\left(c+3\right)} and \frac{8c^{3}\times 4c}{4c\left(c+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{c-8}{c+3}}{\frac{\left(c-2\right)\left(c+3\right)+8c^{3}\times 4c}{4c\left(c+3\right)}}$$

Do the multiplications in \left(c-2\right)\left(c+3\right)+8c^{3}\times 4c.

$$\frac{\frac{c-8}{c+3}}{\frac{c^{2}+3c-2c-6+32c^{4}}{4c\left(c+3\right)}}$$

Combine like terms in c^{2}+3c-2c-6+32c^{4}.

$$\frac{\frac{c-8}{c+3}}{\frac{c^{2}+c-6+32c^{4}}{4c\left(c+3\right)}}$$

Divide \frac{c-8}{c+3} by \frac{c^{2}+c-6+32c^{4}}{4c\left(c+3\right)} by multiplying \frac{c-8}{c+3} by the reciprocal of \frac{c^{2}+c-6+32c^{4}}{4c\left(c+3\right)}.

$$\frac{\left(c-8\right)\times 4c\left(c+3\right)}{\left(c+3\right)\left(c^{2}+c-6+32c^{4}\right)}$$

Cancel out c+3 in both numerator and denominator.

$$\frac{4c\left(c-8\right)}{32c^{4}+c^{2}+c-6}$$

Use the distributive property to multiply 4c by c-8.

$$\frac{4c^{2}-32c}{32c^{4}+c^{2}+c-6}$$

Expand

$$\frac{4\left(c^{2}-8c\right)}{32c^{4}+c^{2}+c-6}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{8c^{4}}{c^{2}+3c}.

$$\frac{\frac{c-8}{c+3}}{\frac{c-2}{4c}+\frac{8c^{4}}{c\left(c+3\right)}}$$

Cancel out c in both numerator and denominator.

$$\frac{\frac{c-8}{c+3}}{\frac{c-2}{4c}+\frac{8c^{3}}{c+3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4c and c+3 is 4c\left(c+3\right). Multiply \frac{c-2}{4c} times \frac{c+3}{c+3}. Multiply \frac{8c^{3}}{c+3} times \frac{4c}{4c}.

$$\frac{\frac{c-8}{c+3}}{\frac{\left(c-2\right)\left(c+3\right)}{4c\left(c+3\right)}+\frac{8c^{3}\times 4c}{4c\left(c+3\right)}}$$

Since \frac{\left(c-2\right)\left(c+3\right)}{4c\left(c+3\right)} and \frac{8c^{3}\times 4c}{4c\left(c+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{c-8}{c+3}}{\frac{\left(c-2\right)\left(c+3\right)+8c^{3}\times 4c}{4c\left(c+3\right)}}$$

Do the multiplications in \left(c-2\right)\left(c+3\right)+8c^{3}\times 4c.

$$\frac{\frac{c-8}{c+3}}{\frac{c^{2}+3c-2c-6+32c^{4}}{4c\left(c+3\right)}}$$

Combine like terms in c^{2}+3c-2c-6+32c^{4}.

$$\frac{\frac{c-8}{c+3}}{\frac{c^{2}+c-6+32c^{4}}{4c\left(c+3\right)}}$$

Divide \frac{c-8}{c+3} by \frac{c^{2}+c-6+32c^{4}}{4c\left(c+3\right)} by multiplying \frac{c-8}{c+3} by the reciprocal of \frac{c^{2}+c-6+32c^{4}}{4c\left(c+3\right)}.

$$\frac{\left(c-8\right)\times 4c\left(c+3\right)}{\left(c+3\right)\left(c^{2}+c-6+32c^{4}\right)}$$

Cancel out c+3 in both numerator and denominator.

$$\frac{4c\left(c-8\right)}{32c^{4}+c^{2}+c-6}$$

Use the distributive property to multiply 4c by c-8.

$$\frac{4c^{2}-32c}{32c^{4}+c^{2}+c-6}$$