Solve a/ab-b^2+b/a^2-ab-a+b/ab

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

Evaluate

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

Short Solution Steps

Factor ab-b^{2}. Factor a^{2}-ab.

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

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

$$\frac{aa}{ab\left(a-b\right)}+\frac{bb}{ab\left(a-b\right)}-\frac{a+b}{ab}$$

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

$$\frac{aa+bb}{ab\left(a-b\right)}-\frac{a+b}{ab}$$

Do the multiplications in aa+bb.

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

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

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

Since \frac{a^{2}+b^{2}}{ab\left(a-b\right)} and \frac{\left(a+b\right)\left(a-b\right)}{ab\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a^{2}+b^{2}-\left(a+b\right)\left(a-b\right).

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

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

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

Cancel out b in both numerator and denominator.

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

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

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

Expand

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

Short Solution Steps

Factor ab-b^{2}. Factor a^{2}-ab.

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

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

$$\frac{aa}{ab\left(a-b\right)}+\frac{bb}{ab\left(a-b\right)}-\frac{a+b}{ab}$$

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

$$\frac{aa+bb}{ab\left(a-b\right)}-\frac{a+b}{ab}$$

Do the multiplications in aa+bb.

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

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

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

Since \frac{a^{2}+b^{2}}{ab\left(a-b\right)} and \frac{\left(a+b\right)\left(a-b\right)}{ab\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a^{2}+b^{2}-\left(a+b\right)\left(a-b\right).

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

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

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

Cancel out b in both numerator and denominator.

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

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

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