Solve a/a-1diva^2-a/a^2-1-1/a-1

$$\frac { a } { a - 1 } \div \frac { a ^ { 2 } - a } { a ^ { 2 } - 1 } - \frac { 1 } { a - 1 }$$

Evaluate

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

Short Solution Steps

Divide \frac{a}{a-1} by \frac{a^{2}-a}{a^{2}-1} by multiplying \frac{a}{a-1} by the reciprocal of \frac{a^{2}-a}{a^{2}-1}.

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

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

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

Cancel out a\left(a-1\right) in both numerator and denominator.

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

Since \frac{a+1}{a-1} and \frac{1}{a-1} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in a+1-1.

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

Expand

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

Short Solution Steps

Divide \frac{a}{a-1} by \frac{a^{2}-a}{a^{2}-1} by multiplying \frac{a}{a-1} by the reciprocal of \frac{a^{2}-a}{a^{2}-1}.

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

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

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

Cancel out a\left(a-1\right) in both numerator and denominator.

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

Since \frac{a+1}{a-1} and \frac{1}{a-1} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in a+1-1.

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