Solve a(a+1)+(a+1)/a^{2-2a+1}*a^2-1/a^2diva^2+a/a^2-a

$$\frac { a ( a + 1 ) + ( a + 1 ) } { a ^ { 2 } - 2 a + 1 } \times \frac { a ^ { 2 } - 1 } { a ^ { 2 } } \div \frac { a ^ { 2 } + a } { a ^ { 2 } - a }$$

Evaluate

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

Short Solution Steps

Multiply \frac{a\left(a+1\right)+a+1}{a^{2}-2a+1} times \frac{a^{2}-1}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{a^{2}+a}{a^{2}-a}.

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

Cancel out a in both numerator and denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out \left(a+1\right)\left(a-1\right)^{2} in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Multiply \frac{a\left(a+1\right)+a+1}{a^{2}-2a+1} times \frac{a^{2}-1}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{a^{2}+a}{a^{2}-a}.

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

Cancel out a in both numerator and denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out \left(a+1\right)\left(a-1\right)^{2} in both numerator and denominator.

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

Expand the expression.

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