Solve (3a+2/a^{3+4a^2+16a}-3a/a^3-64-1/4a-a^2)*(a-8+14a/a-2)^2

$$( \frac { 3 a + 2 } { a ^ { 3 } + 4 a ^ { 2 } + 16 a } - \frac { 3 a } { a ^ { 3 } - 64 } - \frac { 1 } { 4 a - a ^ { 2 } } ) \cdot ( a - 8 + \frac { 14 a } { a - 2 } ) ^ { 2 }$$

Evaluate

$$\frac{a^{2}+4a+16}{a\left(a-2\right)}$$

Short Solution Steps

Factor a^{3}+4a^{2}+16a. Factor a^{3}-64.

$$\left(\frac{3a+2}{a\left(a^{2}+4a+16\right)}-\frac{3a}{\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a^{2}+4a+16\right) and \left(a-4\right)\left(a^{2}+4a+16\right) is a\left(a-4\right)\left(a^{2}+4a+16\right). Multiply \frac{3a+2}{a\left(a^{2}+4a+16\right)} times \frac{a-4}{a-4}. Multiply \frac{3a}{\left(a-4\right)\left(a^{2}+4a+16\right)} times \frac{a}{a}.

$$\left(\frac{\left(3a+2\right)\left(a-4\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{3aa}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Since \frac{\left(3a+2\right)\left(a-4\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)} and \frac{3aa}{a\left(a-4\right)\left(a^{2}+4a+16\right)} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{\left(3a+2\right)\left(a-4\right)-3aa}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Do the multiplications in \left(3a+2\right)\left(a-4\right)-3aa.

$$\left(\frac{3a^{2}-12a+2a-8-3a^{2}}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Combine like terms in 3a^{2}-12a+2a-8-3a^{2}.

$$\left(\frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Factor 4a-a^{2}.

$$\left(\frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{a\left(-a+4\right)}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-4\right)\left(a^{2}+4a+16\right) and a\left(-a+4\right) is a\left(a-4\right)\left(a^{2}+4a+16\right). Multiply \frac{1}{a\left(-a+4\right)} times \frac{-\left(a^{2}+4a+16\right)}{-\left(a^{2}+4a+16\right)}.

$$\left(\frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{-\left(a^{2}+4a+16\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Since \frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)} and \frac{-\left(a^{2}+4a+16\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-10a-8-\left(-\left(a^{2}+4a+16\right)\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Do the multiplications in -10a-8-\left(-\left(a^{2}+4a+16\right)\right).

$$\frac{-10a-8+a^{2}+4a+16}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Combine like terms in -10a-8+a^{2}+4a+16.

$$\frac{-6a+8+a^{2}}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Factor the expressions that are not already factored in \frac{-6a+8+a^{2}}{a\left(a-4\right)\left(a^{2}+4a+16\right)}.

$$\frac{\left(a-4\right)\left(a-2\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Cancel out a-4 in both numerator and denominator.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-8 times \frac{a-2}{a-2}.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\left(\frac{\left(a-8\right)\left(a-2\right)}{a-2}+\frac{14a}{a-2}\right)^{2}$$

Since \frac{\left(a-8\right)\left(a-2\right)}{a-2} and \frac{14a}{a-2} have the same denominator, add them by adding their numerators.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \left(\frac{\left(a-8\right)\left(a-2\right)+14a}{a-2}\right)^{2}$$

Do the multiplications in \left(a-8\right)\left(a-2\right)+14a.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \left(\frac{a^{2}-2a-8a+16+14a}{a-2}\right)^{2}$$

Combine like terms in a^{2}-2a-8a+16+14a.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \left(\frac{a^{2}+4a+16}{a-2}\right)^{2}$$

To raise \frac{a^{2}+4a+16}{a-2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \frac{\left(a^{2}+4a+16\right)^{2}}{\left(a-2\right)^{2}}$$

Multiply \frac{a-2}{a\left(a^{2}+4a+16\right)} times \frac{\left(a^{2}+4a+16\right)^{2}}{\left(a-2\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a-2\right)\left(a^{2}+4a+16\right)^{2}}{a\left(a^{2}+4a+16\right)\left(a-2\right)^{2}}$$

Cancel out \left(a-2\right)\left(a^{2}+4a+16\right) in both numerator and denominator.

$$\frac{a^{2}+4a+16}{a\left(a-2\right)}$$

Use the distributive property to multiply a by a-2.

$$\frac{a^{2}+4a+16}{a^{2}-2a}$$

Expand

$$\frac{a^{2}+4a+16}{a\left(a-2\right)}$$

Short Solution Steps

Factor a^{3}+4a^{2}+16a. Factor a^{3}-64.

$$\left(\frac{3a+2}{a\left(a^{2}+4a+16\right)}-\frac{3a}{\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a^{2}+4a+16\right) and \left(a-4\right)\left(a^{2}+4a+16\right) is a\left(a-4\right)\left(a^{2}+4a+16\right). Multiply \frac{3a+2}{a\left(a^{2}+4a+16\right)} times \frac{a-4}{a-4}. Multiply \frac{3a}{\left(a-4\right)\left(a^{2}+4a+16\right)} times \frac{a}{a}.

$$\left(\frac{\left(3a+2\right)\left(a-4\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{3aa}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Since \frac{\left(3a+2\right)\left(a-4\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)} and \frac{3aa}{a\left(a-4\right)\left(a^{2}+4a+16\right)} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{\left(3a+2\right)\left(a-4\right)-3aa}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Do the multiplications in \left(3a+2\right)\left(a-4\right)-3aa.

$$\left(\frac{3a^{2}-12a+2a-8-3a^{2}}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Combine like terms in 3a^{2}-12a+2a-8-3a^{2}.

$$\left(\frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{4a-a^{2}}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Factor 4a-a^{2}.

$$\left(\frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{1}{a\left(-a+4\right)}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-4\right)\left(a^{2}+4a+16\right) and a\left(-a+4\right) is a\left(a-4\right)\left(a^{2}+4a+16\right). Multiply \frac{1}{a\left(-a+4\right)} times \frac{-\left(a^{2}+4a+16\right)}{-\left(a^{2}+4a+16\right)}.

$$\left(\frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)}-\frac{-\left(a^{2}+4a+16\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\right)\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Since \frac{-10a-8}{a\left(a-4\right)\left(a^{2}+4a+16\right)} and \frac{-\left(a^{2}+4a+16\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-10a-8-\left(-\left(a^{2}+4a+16\right)\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Do the multiplications in -10a-8-\left(-\left(a^{2}+4a+16\right)\right).

$$\frac{-10a-8+a^{2}+4a+16}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Combine like terms in -10a-8+a^{2}+4a+16.

$$\frac{-6a+8+a^{2}}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Factor the expressions that are not already factored in \frac{-6a+8+a^{2}}{a\left(a-4\right)\left(a^{2}+4a+16\right)}.

$$\frac{\left(a-4\right)\left(a-2\right)}{a\left(a-4\right)\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

Cancel out a-4 in both numerator and denominator.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\left(a-8+\frac{14a}{a-2}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-8 times \frac{a-2}{a-2}.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\left(\frac{\left(a-8\right)\left(a-2\right)}{a-2}+\frac{14a}{a-2}\right)^{2}$$

Since \frac{\left(a-8\right)\left(a-2\right)}{a-2} and \frac{14a}{a-2} have the same denominator, add them by adding their numerators.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \left(\frac{\left(a-8\right)\left(a-2\right)+14a}{a-2}\right)^{2}$$

Do the multiplications in \left(a-8\right)\left(a-2\right)+14a.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \left(\frac{a^{2}-2a-8a+16+14a}{a-2}\right)^{2}$$

Combine like terms in a^{2}-2a-8a+16+14a.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \left(\frac{a^{2}+4a+16}{a-2}\right)^{2}$$

To raise \frac{a^{2}+4a+16}{a-2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{a-2}{a\left(a^{2}+4a+16\right)}\times \frac{\left(a^{2}+4a+16\right)^{2}}{\left(a-2\right)^{2}}$$

Multiply \frac{a-2}{a\left(a^{2}+4a+16\right)} times \frac{\left(a^{2}+4a+16\right)^{2}}{\left(a-2\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a-2\right)\left(a^{2}+4a+16\right)^{2}}{a\left(a^{2}+4a+16\right)\left(a-2\right)^{2}}$$

Cancel out \left(a-2\right)\left(a^{2}+4a+16\right) in both numerator and denominator.

$$\frac{a^{2}+4a+16}{a\left(a-2\right)}$$

Use the distributive property to multiply a by a-2.

$$\frac{a^{2}+4a+16}{a^{2}-2a}$$