Solve b/a^{2+ab}-a/b^{2-ab}-a^2+b^2/a^2b-b^3

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

Evaluate

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

Short Solution Steps

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

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

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

$$\frac{bb\left(-a+b\right)}{ab\left(a+b\right)\left(-a+b\right)}-\frac{aa\left(a+b\right)}{ab\left(a+b\right)\left(-a+b\right)}-\frac{a^{2}+b^{2}}{a^{2}b-b^{3}}$$

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

$$\frac{bb\left(-a+b\right)-aa\left(a+b\right)}{ab\left(a+b\right)\left(-a+b\right)}-\frac{a^{2}+b^{2}}{a^{2}b-b^{3}}$$

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

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

Factor a^{2}b-b^{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of ab\left(a+b\right)\left(-a+b\right) and b\left(a+b\right)\left(a-b\right) is ab\left(a+b\right)\left(a-b\right). Multiply \frac{-b^{2}a+b^{3}-a^{3}-a^{2}b}{ab\left(a+b\right)\left(-a+b\right)} times \frac{-1}{-1}. Multiply \frac{a^{2}+b^{2}}{b\left(a+b\right)\left(a-b\right)} times \frac{a}{a}.

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

Since \frac{-\left(-b^{2}a+b^{3}-a^{3}-a^{2}b\right)}{ab\left(a+b\right)\left(a-b\right)} and \frac{\left(a^{2}+b^{2}\right)a}{ab\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(-b^{2}a+b^{3}-a^{3}-a^{2}b\right)-\left(a^{2}+b^{2}\right)a.

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

Combine like terms in b^{2}a-b^{3}+a^{3}+a^{2}b-a^{3}-b^{2}a.

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

Factor the expressions that are not already factored in \frac{a^{2}b-b^{3}}{ab\left(a+b\right)\left(a-b\right)}.

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

Cancel out b\left(a+b\right)\left(a-b\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

$$\frac{bb\left(-a+b\right)}{ab\left(a+b\right)\left(-a+b\right)}-\frac{aa\left(a+b\right)}{ab\left(a+b\right)\left(-a+b\right)}-\frac{a^{2}+b^{2}}{a^{2}b-b^{3}}$$

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

$$\frac{bb\left(-a+b\right)-aa\left(a+b\right)}{ab\left(a+b\right)\left(-a+b\right)}-\frac{a^{2}+b^{2}}{a^{2}b-b^{3}}$$

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

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

Factor a^{2}b-b^{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of ab\left(a+b\right)\left(-a+b\right) and b\left(a+b\right)\left(a-b\right) is ab\left(a+b\right)\left(a-b\right). Multiply \frac{-b^{2}a+b^{3}-a^{3}-a^{2}b}{ab\left(a+b\right)\left(-a+b\right)} times \frac{-1}{-1}. Multiply \frac{a^{2}+b^{2}}{b\left(a+b\right)\left(a-b\right)} times \frac{a}{a}.

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

Since \frac{-\left(-b^{2}a+b^{3}-a^{3}-a^{2}b\right)}{ab\left(a+b\right)\left(a-b\right)} and \frac{\left(a^{2}+b^{2}\right)a}{ab\left(a+b\right)\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(-b^{2}a+b^{3}-a^{3}-a^{2}b\right)-\left(a^{2}+b^{2}\right)a.

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

Combine like terms in b^{2}a-b^{3}+a^{3}+a^{2}b-a^{3}-b^{2}a.

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

Factor the expressions that are not already factored in \frac{a^{2}b-b^{3}}{ab\left(a+b\right)\left(a-b\right)}.

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

Cancel out b\left(a+b\right)\left(a-b\right) in both numerator and denominator.

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