Solve (1-4/a^2-4)diva/a-2

$$( 1 - \frac { 4 } { a ^ { 2 } - 4 } ) \div \frac { a } { a - 2 }$$

Evaluate

$$\frac{a^{2}-8}{a\left(a+2\right)}$$

Short Solution Steps

Factor a^{2}-4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}.

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

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

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

Do the multiplications in \left(a-2\right)\left(a+2\right)-4.

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

Combine like terms in a^{2}+2a-2a-4-4.

$$\frac{\frac{a^{2}-8}{\left(a-2\right)\left(a+2\right)}}{\frac{a}{a-2}}$$

Divide \frac{a^{2}-8}{\left(a-2\right)\left(a+2\right)} by \frac{a}{a-2} by multiplying \frac{a^{2}-8}{\left(a-2\right)\left(a+2\right)} by the reciprocal of \frac{a}{a-2}.

$$\frac{\left(a^{2}-8\right)\left(a-2\right)}{\left(a-2\right)\left(a+2\right)a}$$

Cancel out a-2 in both numerator and denominator.

$$\frac{a^{2}-8}{a\left(a+2\right)}$$

Use the distributive property to multiply a by a+2.

$$\frac{a^{2}-8}{a^{2}+2a}$$

Expand

$$\frac{a^{2}-8}{a\left(a+2\right)}$$

Short Solution Steps

Factor a^{2}-4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(a-2\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}.

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

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

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

Do the multiplications in \left(a-2\right)\left(a+2\right)-4.

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

Combine like terms in a^{2}+2a-2a-4-4.

$$\frac{\frac{a^{2}-8}{\left(a-2\right)\left(a+2\right)}}{\frac{a}{a-2}}$$

Divide \frac{a^{2}-8}{\left(a-2\right)\left(a+2\right)} by \frac{a}{a-2} by multiplying \frac{a^{2}-8}{\left(a-2\right)\left(a+2\right)} by the reciprocal of \frac{a}{a-2}.

$$\frac{\left(a^{2}-8\right)\left(a-2\right)}{\left(a-2\right)\left(a+2\right)a}$$

Cancel out a-2 in both numerator and denominator.

$$\frac{a^{2}-8}{a\left(a+2\right)}$$

Use the distributive property to multiply a by a+2.

$$\frac{a^{2}-8}{a^{2}+2a}$$