Solve 1+a/a-b-1-a/a+b

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-b and a+b is \left(a+b\right)\left(a-b\right). Multiply \frac{1+a}{a-b} times \frac{a+b}{a+b}. Multiply \frac{1-a}{a+b} times \frac{a-b}{a-b}.

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

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

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

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

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

Combine like terms in a+b+a^{2}+ab-a+b+a^{2}-ab.

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

Expand \left(a+b\right)\left(a-b\right).

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-b and a+b is \left(a+b\right)\left(a-b\right). Multiply \frac{1+a}{a-b} times \frac{a+b}{a+b}. Multiply \frac{1-a}{a+b} times \frac{a-b}{a-b}.

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

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

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

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

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

Combine like terms in a+b+a^{2}+ab-a+b+a^{2}-ab.

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

Expand \left(a+b\right)\left(a-b\right).

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