Solve (a+b/ab^{2}-a-b/a^{2b})^{2}*frac{a^{5}b^{6}}{a^{5}b^2-ab^6}-2ab/a^2-b^2

$$( \frac { a + b } { a b ^ { 2 } } - \frac { a - b } { a ^ { 2 } b } ) ^ { 2 } \cdot \frac { a ^ { 5 } b ^ { 6 } } { a ^ { 5 } b ^ { 2 } - a b ^ { 6 } } - \frac { 2 a b } { a ^ { 2 } - b ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

$$\left(\frac{\left(a+b\right)a}{a^{2}b^{2}}-\frac{\left(a-b\right)b}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\left(\frac{\left(a+b\right)a-\left(a-b\right)b}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\left(\frac{a^{2}+ba-ab+b^{2}}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\left(\frac{a^{2}+b^{2}}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

To raise \frac{a^{2}+b^{2}}{a^{2}b^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

Factor the expressions that are not already factored in \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}}\times \frac{a^{5}b^{6}}{a\left(a+b\right)\left(a-b\right)b^{2}\left(a^{2}+b^{2}\right)}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}}\times \frac{a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}+b^{2}\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Multiply \frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}} times \frac{a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}+b^{2}\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}+b^{2}\right)^{2}a^{4}b^{4}}{\left(a^{2}b^{2}\right)^{2}\left(a+b\right)\left(a-b\right)\left(a^{2}+b^{2}\right)}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}-\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

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

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}-\frac{2ab\left(a^{2}b^{2}\right)^{2}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}$$

Since \frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}} and \frac{2ab\left(a^{2}b^{2}\right)^{2}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}-2ab\left(a^{2}b^{2}\right)^{2}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}$$

Expand \left(a^{2}b^{2}\right)^{2}.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}}-\frac{2ab}{a^{2}-b^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)a^{4}\left(b^{2}\right)^{2}}-\frac{2ab}{a^{2}-b^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)a^{4}b^{4}}-\frac{2ab}{a^{2}-b^{2}}$$

Cancel out a^{4}b^{4} in both numerator and denominator.

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

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

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

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

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

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

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

Cancel out a-b in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

$$\left(\frac{\left(a+b\right)a}{a^{2}b^{2}}-\frac{\left(a-b\right)b}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\left(\frac{\left(a+b\right)a-\left(a-b\right)b}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\left(\frac{a^{2}+ba-ab+b^{2}}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\left(\frac{a^{2}+b^{2}}{a^{2}b^{2}}\right)^{2}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

To raise \frac{a^{2}+b^{2}}{a^{2}b^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}}\times \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}-\frac{2ab}{a^{2}-b^{2}}$$

Factor the expressions that are not already factored in \frac{a^{5}b^{6}}{a^{5}b^{2}-ab^{6}}.

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}}\times \frac{a^{5}b^{6}}{a\left(a+b\right)\left(a-b\right)b^{2}\left(a^{2}+b^{2}\right)}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}}\times \frac{a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}+b^{2}\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Multiply \frac{\left(a^{2}+b^{2}\right)^{2}}{\left(a^{2}b^{2}\right)^{2}} times \frac{a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}+b^{2}\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}+b^{2}\right)^{2}a^{4}b^{4}}{\left(a^{2}b^{2}\right)^{2}\left(a+b\right)\left(a-b\right)\left(a^{2}+b^{2}\right)}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}-\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

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

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}-\frac{2ab\left(a^{2}b^{2}\right)^{2}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}$$

Since \frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}} and \frac{2ab\left(a^{2}b^{2}\right)^{2}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}-2ab\left(a^{2}b^{2}\right)^{2}}{\left(a+b\right)\left(a-b\right)\left(a^{2}b^{2}\right)^{2}}$$

Expand \left(a^{2}b^{2}\right)^{2}.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}}-\frac{2ab}{a^{2}-b^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)a^{4}\left(b^{2}\right)^{2}}-\frac{2ab}{a^{2}-b^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(a^{2}+b^{2}\right)a^{4}b^{4}}{\left(a+b\right)\left(a-b\right)a^{4}b^{4}}-\frac{2ab}{a^{2}-b^{2}}$$

Cancel out a^{4}b^{4} in both numerator and denominator.

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

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

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

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

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

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

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

Cancel out a-b in both numerator and denominator.

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