Solve a^2-a/4+1/a-1/frac{4{a-1}+4/a^2}

$$\frac { \frac { a ^ { 2 } - a } { 4 } + \frac { 1 } { a - 1 } } { \frac { 4 } { a - 1 } + \frac { 4 } { a ^ { 2 } } }$$

Evaluate

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

Short Solution Steps

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

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

Since \frac{\left(a^{2}-a\right)\left(a-1\right)}{4\left(a-1\right)} and \frac{4}{4\left(a-1\right)} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in a^{3}-a^{2}-a^{2}+a+4.

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

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

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

Since \frac{4a^{2}}{\left(a-1\right)a^{2}} and \frac{4\left(a-1\right)}{\left(a-1\right)a^{2}} have the same denominator, add them by adding their numerators.

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

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

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

Divide \frac{a^{3}-2a^{2}+a+4}{4\left(a-1\right)} by \frac{4a^{2}+4a-4}{\left(a-1\right)a^{2}} by multiplying \frac{a^{3}-2a^{2}+a+4}{4\left(a-1\right)} by the reciprocal of \frac{4a^{2}+4a-4}{\left(a-1\right)a^{2}}.

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

Cancel out a-1 in both numerator and denominator.

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

Use the distributive property to multiply a^{2} by a^{3}-2a^{2}+a+4.

$$\frac{a^{5}-2a^{4}+a^{3}+4a^{2}}{4\left(4a^{2}+4a-4\right)}$$

Use the distributive property to multiply 4 by 4a^{2}+4a-4.

$$\frac{a^{5}-2a^{4}+a^{3}+4a^{2}}{16a^{2}+16a-16}$$

Expand

$$\frac{a^{5}-2a^{4}+a^{3}+4a^{2}}{16\left(a^{2}+a-1\right)}$$

Short Solution Steps

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

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

Since \frac{\left(a^{2}-a\right)\left(a-1\right)}{4\left(a-1\right)} and \frac{4}{4\left(a-1\right)} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in a^{3}-a^{2}-a^{2}+a+4.

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

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

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

Since \frac{4a^{2}}{\left(a-1\right)a^{2}} and \frac{4\left(a-1\right)}{\left(a-1\right)a^{2}} have the same denominator, add them by adding their numerators.

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

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

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

Divide \frac{a^{3}-2a^{2}+a+4}{4\left(a-1\right)} by \frac{4a^{2}+4a-4}{\left(a-1\right)a^{2}} by multiplying \frac{a^{3}-2a^{2}+a+4}{4\left(a-1\right)} by the reciprocal of \frac{4a^{2}+4a-4}{\left(a-1\right)a^{2}}.

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

Cancel out a-1 in both numerator and denominator.

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

Use the distributive property to multiply a^{2} by a^{3}-2a^{2}+a+4.

$$\frac{a^{5}-2a^{4}+a^{3}+4a^{2}}{4\left(4a^{2}+4a-4\right)}$$

Use the distributive property to multiply 4 by 4a^{2}+4a-4.

$$\frac{a^{5}-2a^{4}+a^{3}+4a^{2}}{16a^{2}+16a-16}$$