Solve a^2-4/a/frac{a^2+4a+4{a^2-4}1/a-2}1/a^2-4a+4

$$\frac{ \frac{ { a }^{ 2 } -4 }{ a } }{ \frac{ { a }^{ 2 } +4a+4 }{ { a }^{ 2 } -4 } \frac{ 1 }{ a-2 } } \frac{ 1 }{ { a }^{ 2 } -4a+4 }$$

Evaluate

$$\frac{a-2}{a}$$

Short Solution Steps

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

$$\frac{\frac{a^{2}-4}{a}}{\frac{\left(a+2\right)^{2}}{\left(a-2\right)\left(a+2\right)}\times \frac{1}{a-2}}\times \frac{1}{a^{2}-4a+4}$$

Cancel out a+2 in both numerator and denominator.

$$\frac{\frac{a^{2}-4}{a}}{\frac{a+2}{a-2}\times \frac{1}{a-2}}\times \frac{1}{a^{2}-4a+4}$$

Multiply \frac{a+2}{a-2} times \frac{1}{a-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{a^{2}-4}{a}}{\frac{a+2}{\left(a-2\right)\left(a-2\right)}}\times \frac{1}{a^{2}-4a+4}$$

Divide \frac{a^{2}-4}{a} by \frac{a+2}{\left(a-2\right)\left(a-2\right)} by multiplying \frac{a^{2}-4}{a} by the reciprocal of \frac{a+2}{\left(a-2\right)\left(a-2\right)}.

$$\frac{\left(a^{2}-4\right)\left(a-2\right)\left(a-2\right)}{a\left(a+2\right)}\times \frac{1}{a^{2}-4a+4}$$

Factor the expressions that are not already factored in \frac{\left(a^{2}-4\right)\left(a-2\right)\left(a-2\right)}{a\left(a+2\right)}.

$$\frac{\left(a+2\right)\left(a-2\right)^{3}}{a\left(a+2\right)}\times \frac{1}{a^{2}-4a+4}$$

Cancel out a+2 in both numerator and denominator.

$$\frac{\left(a-2\right)^{3}}{a}\times \frac{1}{a^{2}-4a+4}$$

Multiply \frac{\left(a-2\right)^{3}}{a} times \frac{1}{a^{2}-4a+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a-2\right)^{3}}{a\left(a^{2}-4a+4\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(a-2\right)^{2} in both numerator and denominator.

$$\frac{a-2}{a}$$

Expand

$$\frac{a-2}{a}$$

Short Solution Steps

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

$$\frac{\frac{a^{2}-4}{a}}{\frac{\left(a+2\right)^{2}}{\left(a-2\right)\left(a+2\right)}\times \frac{1}{a-2}}\times \frac{1}{a^{2}-4a+4}$$

Cancel out a+2 in both numerator and denominator.

$$\frac{\frac{a^{2}-4}{a}}{\frac{a+2}{a-2}\times \frac{1}{a-2}}\times \frac{1}{a^{2}-4a+4}$$

Multiply \frac{a+2}{a-2} times \frac{1}{a-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{a^{2}-4}{a}}{\frac{a+2}{\left(a-2\right)\left(a-2\right)}}\times \frac{1}{a^{2}-4a+4}$$

Divide \frac{a^{2}-4}{a} by \frac{a+2}{\left(a-2\right)\left(a-2\right)} by multiplying \frac{a^{2}-4}{a} by the reciprocal of \frac{a+2}{\left(a-2\right)\left(a-2\right)}.

$$\frac{\left(a^{2}-4\right)\left(a-2\right)\left(a-2\right)}{a\left(a+2\right)}\times \frac{1}{a^{2}-4a+4}$$

Factor the expressions that are not already factored in \frac{\left(a^{2}-4\right)\left(a-2\right)\left(a-2\right)}{a\left(a+2\right)}.

$$\frac{\left(a+2\right)\left(a-2\right)^{3}}{a\left(a+2\right)}\times \frac{1}{a^{2}-4a+4}$$

Cancel out a+2 in both numerator and denominator.

$$\frac{\left(a-2\right)^{3}}{a}\times \frac{1}{a^{2}-4a+4}$$

Multiply \frac{\left(a-2\right)^{3}}{a} times \frac{1}{a^{2}-4a+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a-2\right)^{3}}{a\left(a^{2}-4a+4\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(a-2\right)^{2} in both numerator and denominator.

$$\frac{a-2}{a}$$