Solve frac{frac{k^{2}-10}{4+k^{2}}-1-4k^2/1+4k^2}{k^2+4-8k/11+4k^2}

$$\frac { \frac { k ^ { 2 } - 10 } { 4 + k ^ { 2 } } - \frac { 1 - 4 k ^ { 2 } } { 1 + 4 k ^ { 2 } } } { k ^ { 2 } + 4 - \frac { 8 k } { 11 + 4 k ^ { 2 } } }$$

Evaluate

$$\frac{2\left(2k^{2}-7\right)\left(2k^{2}+1\right)\left(4k^{2}+11\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(4k^{4}+27k^{2}-8k+44\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4+k^{2} and 1+4k^{2} is \left(4k^{2}+1\right)\left(k^{2}+4\right). Multiply \frac{k^{2}-10}{4+k^{2}} times \frac{4k^{2}+1}{4k^{2}+1}. Multiply \frac{1-4k^{2}}{1+4k^{2}} times \frac{k^{2}+4}{k^{2}+4}.

$$\frac{\frac{\left(k^{2}-10\right)\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}-\frac{\left(1-4k^{2}\right)\left(k^{2}+4\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

Since \frac{\left(k^{2}-10\right)\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} and \frac{\left(1-4k^{2}\right)\left(k^{2}+4\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(k^{2}-10\right)\left(4k^{2}+1\right)-\left(1-4k^{2}\right)\left(k^{2}+4\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

Do the multiplications in \left(k^{2}-10\right)\left(4k^{2}+1\right)-\left(1-4k^{2}\right)\left(k^{2}+4\right).

$$\frac{\frac{4k^{4}+k^{2}-40k^{2}-10-k^{2}-4+4k^{4}+16k^{2}}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

Combine like terms in 4k^{4}+k^{2}-40k^{2}-10-k^{2}-4+4k^{4}+16k^{2}.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply k^{2}+4 times \frac{11+4k^{2}}{11+4k^{2}}.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{\left(k^{2}+4\right)\left(11+4k^{2}\right)}{11+4k^{2}}-\frac{8k}{11+4k^{2}}}$$

Since \frac{\left(k^{2}+4\right)\left(11+4k^{2}\right)}{11+4k^{2}} and \frac{8k}{11+4k^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{\left(k^{2}+4\right)\left(11+4k^{2}\right)-8k}{11+4k^{2}}}$$

Do the multiplications in \left(k^{2}+4\right)\left(11+4k^{2}\right)-8k.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{11k^{2}+4k^{4}+44+16k^{2}-8k}{11+4k^{2}}}$$

Combine like terms in 11k^{2}+4k^{4}+44+16k^{2}-8k.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{27k^{2}+4k^{4}+44-8k}{11+4k^{2}}}$$

Divide \frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} by \frac{27k^{2}+4k^{4}+44-8k}{11+4k^{2}} by multiplying \frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} by the reciprocal of \frac{27k^{2}+4k^{4}+44-8k}{11+4k^{2}}.

$$\frac{\left(8k^{4}-24k^{2}-14\right)\left(11+4k^{2}\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(27k^{2}+4k^{4}+44-8k\right)}$$

Use the distributive property to multiply 8k^{4}-24k^{2}-14 by 11+4k^{2} and combine like terms.

$$\frac{-8k^{4}+32k^{6}-320k^{2}-154}{\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(27k^{2}+4k^{4}+44-8k\right)}$$

Use the distributive property to multiply 4k^{2}+1 by k^{2}+4 and combine like terms.

$$\frac{-8k^{4}+32k^{6}-320k^{2}-154}{\left(4k^{4}+17k^{2}+4\right)\left(27k^{2}+4k^{4}+44-8k\right)}$$

Use the distributive property to multiply 4k^{4}+17k^{2}+4 by 27k^{2}+4k^{4}+44-8k and combine like terms.

$$\frac{-8k^{4}+32k^{6}-320k^{2}-154}{176k^{6}+16k^{8}+651k^{4}-32k^{5}+856k^{2}-136k^{3}+176-32k}$$

Factor the expressions that are not already factored.

$$\frac{2\left(2k^{2}-7\right)\left(2k^{2}+1\right)\left(4k^{2}+11\right)}{16\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(\frac{1}{4}k^{4}+\frac{27}{16}k^{2}-\frac{1}{2}k+\frac{11}{4}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(2k^{2}-7\right)\left(2k^{2}+1\right)\left(4k^{2}+11\right)}{8\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(\frac{1}{4}k^{4}+\frac{27}{16}k^{2}-\frac{1}{2}k+\frac{11}{4}\right)}$$

Expand the expression.

$$\frac{16k^{6}-4k^{4}-160k^{2}-77}{8k^{8}+88k^{6}-16k^{5}+\frac{651}{2}k^{4}-68k^{3}+428k^{2}-16k+88}$$

Expand

$$\frac{2\left(16k^{6}-4k^{4}-160k^{2}-77\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(4k^{4}+27k^{2}-8k+44\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4+k^{2} and 1+4k^{2} is \left(4k^{2}+1\right)\left(k^{2}+4\right). Multiply \frac{k^{2}-10}{4+k^{2}} times \frac{4k^{2}+1}{4k^{2}+1}. Multiply \frac{1-4k^{2}}{1+4k^{2}} times \frac{k^{2}+4}{k^{2}+4}.

$$\frac{\frac{\left(k^{2}-10\right)\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}-\frac{\left(1-4k^{2}\right)\left(k^{2}+4\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

Since \frac{\left(k^{2}-10\right)\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} and \frac{\left(1-4k^{2}\right)\left(k^{2}+4\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(k^{2}-10\right)\left(4k^{2}+1\right)-\left(1-4k^{2}\right)\left(k^{2}+4\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

Do the multiplications in \left(k^{2}-10\right)\left(4k^{2}+1\right)-\left(1-4k^{2}\right)\left(k^{2}+4\right).

$$\frac{\frac{4k^{4}+k^{2}-40k^{2}-10-k^{2}-4+4k^{4}+16k^{2}}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

Combine like terms in 4k^{4}+k^{2}-40k^{2}-10-k^{2}-4+4k^{4}+16k^{2}.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{k^{2}+4-\frac{8k}{11+4k^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply k^{2}+4 times \frac{11+4k^{2}}{11+4k^{2}}.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{\left(k^{2}+4\right)\left(11+4k^{2}\right)}{11+4k^{2}}-\frac{8k}{11+4k^{2}}}$$

Since \frac{\left(k^{2}+4\right)\left(11+4k^{2}\right)}{11+4k^{2}} and \frac{8k}{11+4k^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{\left(k^{2}+4\right)\left(11+4k^{2}\right)-8k}{11+4k^{2}}}$$

Do the multiplications in \left(k^{2}+4\right)\left(11+4k^{2}\right)-8k.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{11k^{2}+4k^{4}+44+16k^{2}-8k}{11+4k^{2}}}$$

Combine like terms in 11k^{2}+4k^{4}+44+16k^{2}-8k.

$$\frac{\frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}}{\frac{27k^{2}+4k^{4}+44-8k}{11+4k^{2}}}$$

Divide \frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} by \frac{27k^{2}+4k^{4}+44-8k}{11+4k^{2}} by multiplying \frac{8k^{4}-24k^{2}-14}{\left(4k^{2}+1\right)\left(k^{2}+4\right)} by the reciprocal of \frac{27k^{2}+4k^{4}+44-8k}{11+4k^{2}}.

$$\frac{\left(8k^{4}-24k^{2}-14\right)\left(11+4k^{2}\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(27k^{2}+4k^{4}+44-8k\right)}$$

Use the distributive property to multiply 8k^{4}-24k^{2}-14 by 11+4k^{2} and combine like terms.

$$\frac{-8k^{4}+32k^{6}-320k^{2}-154}{\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(27k^{2}+4k^{4}+44-8k\right)}$$

Use the distributive property to multiply 4k^{2}+1 by k^{2}+4 and combine like terms.

$$\frac{-8k^{4}+32k^{6}-320k^{2}-154}{\left(4k^{4}+17k^{2}+4\right)\left(27k^{2}+4k^{4}+44-8k\right)}$$

Use the distributive property to multiply 4k^{4}+17k^{2}+4 by 27k^{2}+4k^{4}+44-8k and combine like terms.

$$\frac{-8k^{4}+32k^{6}-320k^{2}-154}{176k^{6}+16k^{8}+651k^{4}-32k^{5}+856k^{2}-136k^{3}+176-32k}$$

Factor the expressions that are not already factored.

$$\frac{2\left(2k^{2}-7\right)\left(2k^{2}+1\right)\left(4k^{2}+11\right)}{16\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(\frac{1}{4}k^{4}+\frac{27}{16}k^{2}-\frac{1}{2}k+\frac{11}{4}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(2k^{2}-7\right)\left(2k^{2}+1\right)\left(4k^{2}+11\right)}{8\left(4k^{2}+1\right)\left(k^{2}+4\right)\left(\frac{1}{4}k^{4}+\frac{27}{16}k^{2}-\frac{1}{2}k+\frac{11}{4}\right)}$$

Expand the expression.

$$\frac{16k^{6}-4k^{4}-160k^{2}-77}{8k^{8}+88k^{6}-16k^{5}+\frac{651}{2}k^{4}-68k^{3}+428k^{2}-16k+88}$$