Solve 1/2a^{3}quad(2)frac{a^{2}-4b^{2}}{a^{2}+4ab+4b^{2}}diva^2+ab-6b^2/a^2+5ab+6b^2

$$\frac { 1 } { 2 a ^ { 3 } } \quad ( 2 ) \frac { a ^ { 2 } - 4 b ^ { 2 } } { a ^ { 2 } + 4 a b + 4 b ^ { 2 } } \div \frac { a ^ { 2 } + a b - 6 b ^ { 2 } } { a ^ { 2 } + 5 a b + 6 b ^ { 2 } }$$

Evaluate

$$\frac{1}{a^{3}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}-4b^{2}}{a^{2}+4ab+4b^{2}}.

$$\frac{\frac{1}{2a^{3}}\times 2\times \frac{\left(a-2b\right)\left(a+2b\right)}{\left(a+2b\right)^{2}}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

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

$$\frac{\frac{1}{2a^{3}}\times 2\times \frac{a-2b}{a+2b}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

Express \frac{1}{2a^{3}}\times 2 as a single fraction.

$$\frac{\frac{2}{2a^{3}}\times \frac{a-2b}{a+2b}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

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

$$\frac{\frac{2\left(a-2b\right)}{2a^{3}\left(a+2b\right)}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\frac{a-2b}{\left(a+2b\right)a^{3}}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

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

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

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

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

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

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

Cancel out \left(a-2b\right)\left(a+2b\right) in both numerator and denominator.

$$\frac{1}{a^{3}}$$

Expand

$$\frac{1}{a^{3}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}-4b^{2}}{a^{2}+4ab+4b^{2}}.

$$\frac{\frac{1}{2a^{3}}\times 2\times \frac{\left(a-2b\right)\left(a+2b\right)}{\left(a+2b\right)^{2}}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

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

$$\frac{\frac{1}{2a^{3}}\times 2\times \frac{a-2b}{a+2b}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

Express \frac{1}{2a^{3}}\times 2 as a single fraction.

$$\frac{\frac{2}{2a^{3}}\times \frac{a-2b}{a+2b}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

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

$$\frac{\frac{2\left(a-2b\right)}{2a^{3}\left(a+2b\right)}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\frac{a-2b}{\left(a+2b\right)a^{3}}}{\frac{a^{2}+ab-6b^{2}}{a^{2}+5ab+6b^{2}}}$$

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

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

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

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

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

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

Cancel out \left(a-2b\right)\left(a+2b\right) in both numerator and denominator.

$$\frac{1}{a^{3}}$$