Solve (a+1/a^2):(a-1+1/a)

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a^{2}}{a^{2}}.

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

Since \frac{aa^{2}}{a^{2}} and \frac{1}{a^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in aa^{2}+1.

$$\frac{\frac{a^{3}+1}{a^{2}}}{a-1+\frac{1}{a}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-1 times \frac{a}{a}.

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

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

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

Do the multiplications in \left(a-1\right)a+1.

$$\frac{\frac{a^{3}+1}{a^{2}}}{\frac{a^{2}-a+1}{a}}$$

Divide \frac{a^{3}+1}{a^{2}} by \frac{a^{2}-a+1}{a} by multiplying \frac{a^{3}+1}{a^{2}} by the reciprocal of \frac{a^{2}-a+1}{a}.

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

Cancel out a in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a^{2}}{a^{2}}.

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

Since \frac{aa^{2}}{a^{2}} and \frac{1}{a^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in aa^{2}+1.

$$\frac{\frac{a^{3}+1}{a^{2}}}{a-1+\frac{1}{a}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-1 times \frac{a}{a}.

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

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

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

Do the multiplications in \left(a-1\right)a+1.

$$\frac{\frac{a^{3}+1}{a^{2}}}{\frac{a^{2}-a+1}{a}}$$

Divide \frac{a^{3}+1}{a^{2}} by \frac{a^{2}-a+1}{a} by multiplying \frac{a^{3}+1}{a^{2}} by the reciprocal of \frac{a^{2}-a+1}{a}.

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

Cancel out a in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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