Solve a-1/adiv(a-1/a)

$$\frac { a - 1 } { a } \div ( a - \frac { 1 } { a } )$$

Evaluate

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

Short Solution Steps

Express \frac{\frac{a-1}{a}}{a-\frac{1}{a}} as a single fraction.

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

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

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

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

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

Do the multiplications in aa-1.

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

Cancel out a and a.

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

Factor the expressions that are not already factored.

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

Cancel out a-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Express \frac{\frac{a-1}{a}}{a-\frac{1}{a}} as a single fraction.

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

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

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

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

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

Do the multiplications in aa-1.

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

Cancel out a and a.

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

Factor the expressions that are not already factored.

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

Cancel out a-1 in both numerator and denominator.

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