Solve frac{a^{2}-a}{a^{2}+1}+2a^2/a^3-a^2+a-1)*(1-1/a^2)=

$$\frac { a ^ { 2 } - a } { a ^ { 2 } + 1 } + \frac { 2 a ^ { 2 } } { a ^ { 3 } - a ^ { 2 } + a - 1 } ) \cdot ( 1 - \frac { 1 } { a ^ { 2 } } ) =$$

Evaluate

$$\frac{a}{a-1}$$

Short Solution Steps

Factor a^{3}-a^{2}+a-1.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{a^{3}+a}{\left(a-1\right)\left(a^{2}+1\right)}.

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

Cancel out a^{2}+1 in both numerator and denominator.

$$\frac{a}{a-1}$$

Expand

$$\frac{a}{a-1}$$

Short Solution Steps

Factor a^{3}-a^{2}+a-1.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{a^{3}+a}{\left(a-1\right)\left(a^{2}+1\right)}.

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

Cancel out a^{2}+1 in both numerator and denominator.

$$\frac{a}{a-1}$$