Solve -27k^3-8k/2(9k^2+4)+271/k^3-81/k/2(9frac{1{k^2}+4)}

$$\frac { - 27 k ^ { 3 } - 8 k } { 2 ( 9 k ^ { 2 } + 4 ) } + \frac { 27 \frac { 1 } { k ^ { 3 } } - 8 \frac { 1 } { k } } { 2 ( 9 \frac { 1 } { k ^ { 2 } } + 4 ) }$$

Evaluate

$$\frac{108+139k^{2}-347k^{4}-108k^{6}}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Short Solution Steps

Use the distributive property to multiply 2 by 9k^{2}+4.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{27\times \frac{1}{k^{3}}-8\times \frac{1}{k}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Express 27\times \frac{1}{k^{3}} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27}{k^{3}}-8\times \frac{1}{k}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Express 8\times \frac{1}{k} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27}{k^{3}}-\frac{8}{k}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of k^{3} and k is k^{3}. Multiply \frac{8}{k} times \frac{k^{2}}{k^{2}}.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27}{k^{3}}-\frac{8k^{2}}{k^{3}}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Since \frac{27}{k^{3}} and \frac{8k^{2}}{k^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Express 9\times \frac{1}{k^{2}} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\left(\frac{9}{k^{2}}+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{k^{2}}{k^{2}}.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\left(\frac{9}{k^{2}}+\frac{4k^{2}}{k^{2}}\right)}$$

Since \frac{9}{k^{2}} and \frac{4k^{2}}{k^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\times \frac{9+4k^{2}}{k^{2}}}$$

Express 2\times \frac{9+4k^{2}}{k^{2}} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{\frac{2\left(9+4k^{2}\right)}{k^{2}}}$$

Divide \frac{27-8k^{2}}{k^{3}} by \frac{2\left(9+4k^{2}\right)}{k^{2}} by multiplying \frac{27-8k^{2}}{k^{3}} by the reciprocal of \frac{2\left(9+4k^{2}\right)}{k^{2}}.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\left(27-8k^{2}\right)k^{2}}{k^{3}\times 2\left(9+4k^{2}\right)}$$

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

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{-8k^{2}+27}{2k\left(4k^{2}+9\right)}$$

Factor 18k^{2}+8.

$$\frac{-27k^{3}-8k}{2\left(9k^{2}+4\right)}+\frac{-8k^{2}+27}{2k\left(4k^{2}+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(9k^{2}+4\right) and 2k\left(4k^{2}+9\right) is 2k\left(4k^{2}+9\right)\left(9k^{2}+4\right). Multiply \frac{-27k^{3}-8k}{2\left(9k^{2}+4\right)} times \frac{k\left(4k^{2}+9\right)}{k\left(4k^{2}+9\right)}. Multiply \frac{-8k^{2}+27}{2k\left(4k^{2}+9\right)} times \frac{9k^{2}+4}{9k^{2}+4}.

$$\frac{\left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}+\frac{\left(-8k^{2}+27\right)\left(9k^{2}+4\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Since \frac{\left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)} and \frac{\left(-8k^{2}+27\right)\left(9k^{2}+4\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)+\left(-8k^{2}+27\right)\left(9k^{2}+4\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Do the multiplications in \left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)+\left(-8k^{2}+27\right)\left(9k^{2}+4\right).

$$\frac{-108k^{6}-243k^{4}-32k^{4}-72k^{2}-72k^{4}-32k^{2}+243k^{2}+108}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Combine like terms in -108k^{6}-243k^{4}-32k^{4}-72k^{2}-72k^{4}-32k^{2}+243k^{2}+108.

$$\frac{-108k^{6}-347k^{4}+139k^{2}+108}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Expand 2k\left(4k^{2}+9\right)\left(9k^{2}+4\right).

$$\frac{-108k^{6}-347k^{4}+139k^{2}+108}{72k^{5}+194k^{3}+72k}$$

Expand

$$-\frac{108k^{6}+347k^{4}-139k^{2}-108}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Short Solution Steps

Use the distributive property to multiply 2 by 9k^{2}+4.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{27\times \frac{1}{k^{3}}-8\times \frac{1}{k}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Express 27\times \frac{1}{k^{3}} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27}{k^{3}}-8\times \frac{1}{k}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Express 8\times \frac{1}{k} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27}{k^{3}}-\frac{8}{k}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of k^{3} and k is k^{3}. Multiply \frac{8}{k} times \frac{k^{2}}{k^{2}}.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27}{k^{3}}-\frac{8k^{2}}{k^{3}}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Since \frac{27}{k^{3}} and \frac{8k^{2}}{k^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\left(9\times \frac{1}{k^{2}}+4\right)}$$

Express 9\times \frac{1}{k^{2}} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\left(\frac{9}{k^{2}}+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{k^{2}}{k^{2}}.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\left(\frac{9}{k^{2}}+\frac{4k^{2}}{k^{2}}\right)}$$

Since \frac{9}{k^{2}} and \frac{4k^{2}}{k^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{2\times \frac{9+4k^{2}}{k^{2}}}$$

Express 2\times \frac{9+4k^{2}}{k^{2}} as a single fraction.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\frac{27-8k^{2}}{k^{3}}}{\frac{2\left(9+4k^{2}\right)}{k^{2}}}$$

Divide \frac{27-8k^{2}}{k^{3}} by \frac{2\left(9+4k^{2}\right)}{k^{2}} by multiplying \frac{27-8k^{2}}{k^{3}} by the reciprocal of \frac{2\left(9+4k^{2}\right)}{k^{2}}.

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{\left(27-8k^{2}\right)k^{2}}{k^{3}\times 2\left(9+4k^{2}\right)}$$

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

$$\frac{-27k^{3}-8k}{18k^{2}+8}+\frac{-8k^{2}+27}{2k\left(4k^{2}+9\right)}$$

Factor 18k^{2}+8.

$$\frac{-27k^{3}-8k}{2\left(9k^{2}+4\right)}+\frac{-8k^{2}+27}{2k\left(4k^{2}+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(9k^{2}+4\right) and 2k\left(4k^{2}+9\right) is 2k\left(4k^{2}+9\right)\left(9k^{2}+4\right). Multiply \frac{-27k^{3}-8k}{2\left(9k^{2}+4\right)} times \frac{k\left(4k^{2}+9\right)}{k\left(4k^{2}+9\right)}. Multiply \frac{-8k^{2}+27}{2k\left(4k^{2}+9\right)} times \frac{9k^{2}+4}{9k^{2}+4}.

$$\frac{\left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}+\frac{\left(-8k^{2}+27\right)\left(9k^{2}+4\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Since \frac{\left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)} and \frac{\left(-8k^{2}+27\right)\left(9k^{2}+4\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)+\left(-8k^{2}+27\right)\left(9k^{2}+4\right)}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Do the multiplications in \left(-27k^{3}-8k\right)k\left(4k^{2}+9\right)+\left(-8k^{2}+27\right)\left(9k^{2}+4\right).

$$\frac{-108k^{6}-243k^{4}-32k^{4}-72k^{2}-72k^{4}-32k^{2}+243k^{2}+108}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Combine like terms in -108k^{6}-243k^{4}-32k^{4}-72k^{2}-72k^{4}-32k^{2}+243k^{2}+108.

$$\frac{-108k^{6}-347k^{4}+139k^{2}+108}{2k\left(4k^{2}+9\right)\left(9k^{2}+4\right)}$$

Expand 2k\left(4k^{2}+9\right)\left(9k^{2}+4\right).

$$\frac{-108k^{6}-347k^{4}+139k^{2}+108}{72k^{5}+194k^{3}+72k}$$