Solve frac{a^{2}-4a+4}{a^{2}+4ab+4b^2}:a^2-4/a^2-4b^2

$$\frac { a ^ { 2 } - 4 a + 4 } { a ^ { 2 } + 4 a b + 4 b ^ { 2 } } : \frac { a ^ { 2 } - 4 } { a ^ { 2 } - 4 b ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Divide \frac{a^{2}-4a+4}{a^{2}+4ab+4b^{2}} by \frac{a^{2}-4}{a^{2}-4b^{2}} by multiplying \frac{a^{2}-4a+4}{a^{2}+4ab+4b^{2}} by the reciprocal of \frac{a^{2}-4}{a^{2}-4b^{2}}.

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

$$\frac{a^{2}-2ab-2a+4b}{a^{2}+2ab+2a+4b}$$

Expand

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

Short Solution Steps

Divide \frac{a^{2}-4a+4}{a^{2}+4ab+4b^{2}} by \frac{a^{2}-4}{a^{2}-4b^{2}} by multiplying \frac{a^{2}-4a+4}{a^{2}+4ab+4b^{2}} by the reciprocal of \frac{a^{2}-4}{a^{2}-4b^{2}}.

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

$$\frac{a^{2}-2ab-2a+4b}{a^{2}+2ab+2a+4b}$$