Solve a+3b/ab-b^{2}:[(a-3b/ab+3b^{2}-frac{2a^{2}+18b^{2}}{a^{2}b-9b^{3}}):a^2+2ab-3b^2/a^2-6ab+9b^2-6/a-b]

$$\frac { a + 3 b } { a b - b ^ { 2 } } : [ ( \frac { a - 3 b } { a b + 3 b ^ { 2 } } - \frac { 2 a ^ { 2 } + 18 b ^ { 2 } } { a ^ { 2 } b - 9 b ^ { 3 } } ) : \frac { a ^ { 2 } + 2 a b - 3 b ^ { 2 } } { a ^ { 2 } - 6 a b + 9 b ^ { 2 } } - \frac { 6 } { a - b } ]$$

Evaluate

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Short Solution Steps

Factor ab+3b^{2}. Factor a^{2}b-9b^{3}.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{a-3b}{b\left(a+3b\right)}-\frac{2a^{2}+18b^{2}}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

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

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{\left(a-3b\right)\left(a-3b\right)}{b\left(a-3b\right)\left(a+3b\right)}-\frac{2a^{2}+18b^{2}}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Since \frac{\left(a-3b\right)\left(a-3b\right)}{b\left(a-3b\right)\left(a+3b\right)} and \frac{2a^{2}+18b^{2}}{b\left(a-3b\right)\left(a+3b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{\left(a-3b\right)\left(a-3b\right)-\left(2a^{2}+18b^{2}\right)}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Do the multiplications in \left(a-3b\right)\left(a-3b\right)-\left(2a^{2}+18b^{2}\right).

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{a^{2}-3ab-3ab+9b^{2}-2a^{2}-18b^{2}}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Combine like terms in a^{2}-3ab-3ab+9b^{2}-2a^{2}-18b^{2}.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{-a^{2}-9b^{2}-6ab}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Factor the expressions that are not already factored in \frac{-a^{2}-9b^{2}-6ab}{b\left(a-3b\right)\left(a+3b\right)}.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{\left(a+3b\right)\left(-a-3b\right)}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Extract the negative sign in -a-3b.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{-\left(a+3b\right)\left(a+3b\right)}{b\left(a-3b\right)\left(a+3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Cancel out a+3b in both numerator and denominator.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{\frac{-\left(a+3b\right)}{b\left(a-3b\right)}}{\frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}}-\frac{6}{a-b}}$$

Divide \frac{-\left(a+3b\right)}{b\left(a-3b\right)} by \frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}} by multiplying \frac{-\left(a+3b\right)}{b\left(a-3b\right)} by the reciprocal of \frac{a^{2}+2ab-3b^{2}}{a^{2}-6ab+9b^{2}}.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{-\left(a+3b\right)\left(a^{2}-6ab+9b^{2}\right)}{b\left(a-3b\right)\left(a^{2}+2ab-3b^{2}\right)}-\frac{6}{a-b}}$$

Factor the expressions that are not already factored in \frac{-\left(a+3b\right)\left(a^{2}-6ab+9b^{2}\right)}{b\left(a-3b\right)\left(a^{2}+2ab-3b^{2}\right)}.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{-\left(a+3b\right)\left(a-3b\right)^{2}}{b\left(a-3b\right)\left(a-b\right)\left(a+3b\right)}-\frac{6}{a-b}}$$

Cancel out \left(a-3b\right)\left(a+3b\right) in both numerator and denominator.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{-\left(a-3b\right)}{b\left(a-b\right)}-\frac{6}{a-b}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b\left(a-b\right) and a-b is b\left(a-b\right). Multiply \frac{6}{a-b} times \frac{b}{b}.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{-\left(a-3b\right)}{b\left(a-b\right)}-\frac{6b}{b\left(a-b\right)}}$$

Since \frac{-\left(a-3b\right)}{b\left(a-b\right)} and \frac{6b}{b\left(a-b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{-\left(a-3b\right)-6b}{b\left(a-b\right)}}$$

Do the multiplications in -\left(a-3b\right)-6b.

$$\frac{\frac{a+3b}{ab-b^{2}}}{\frac{-a+3b-6b}{b\left(a-b\right)}}$$

Combine like terms in -a+3b-6b.

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

Divide \frac{a+3b}{ab-b^{2}} by \frac{-a-3b}{b\left(a-b\right)} by multiplying \frac{a+3b}{ab-b^{2}} by the reciprocal of \frac{-a-3b}{b\left(a-b\right)}.

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

Extract the negative sign in a+3b.

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

Cancel out -a-3b in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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Factor

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