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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Since \frac{a-1}{b-1} and \frac{1}{b-1} have the same denominator, add them by adding their numerators.

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

Combine like terms in a-1+1.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Since \frac{a-1}{b-1} and \frac{1}{b-1} have the same denominator, add them by adding their numerators.

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

Combine like terms in a-1+1.

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