Solve frac{a/a^{3-b^{3}}*a^2+b^2+ab/a+b-1/b-a}{2a+b/a^2+2ab+b^2}

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

Evaluate

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

Short Solution Steps

Multiply \frac{a}{a^{3}-b^{3}} times \frac{a^{2}+b^{2}+ab}{a+b} by multiplying numerator times numerator and denominator times denominator.

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

Factor \left(a^{3}-b^{3}\right)\left(a+b\right).

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -a^{2}-ab-b^{2}.

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

Cancel out a^{2}+ab+b^{2} in both numerator and denominator.

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

Divide \frac{-\left(2a+b\right)}{\left(a-b\right)\left(-a-b\right)} by \frac{2a+b}{a^{2}+2ab+b^{2}} by multiplying \frac{-\left(2a+b\right)}{\left(a-b\right)\left(-a-b\right)} by the reciprocal of \frac{2a+b}{a^{2}+2ab+b^{2}}.

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

Cancel out 2a+b in both numerator and denominator.

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

To find the opposite of a^{2}+2ab+b^{2}, find the opposite of each term.

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

Use the distributive property to multiply a-b by -a-b and combine like terms.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in a+b.

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

Cancel out -a-b in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Multiply \frac{a}{a^{3}-b^{3}} times \frac{a^{2}+b^{2}+ab}{a+b} by multiplying numerator times numerator and denominator times denominator.

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

Factor \left(a^{3}-b^{3}\right)\left(a+b\right).

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -a^{2}-ab-b^{2}.

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

Cancel out a^{2}+ab+b^{2} in both numerator and denominator.

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

Divide \frac{-\left(2a+b\right)}{\left(a-b\right)\left(-a-b\right)} by \frac{2a+b}{a^{2}+2ab+b^{2}} by multiplying \frac{-\left(2a+b\right)}{\left(a-b\right)\left(-a-b\right)} by the reciprocal of \frac{2a+b}{a^{2}+2ab+b^{2}}.

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

Cancel out 2a+b in both numerator and denominator.

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

To find the opposite of a^{2}+2ab+b^{2}, find the opposite of each term.

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

Use the distributive property to multiply a-b by -a-b and combine like terms.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in a+b.

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

Cancel out -a-b in both numerator and denominator.

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

Expand the expression.

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