Solve (a-2/a+2+8a/a^2-4)diva+2/a^2-2a

$$( \frac { a - 2 } { a + 2 } + \frac { 8 a } { a ^ { 2 } - 4 } ) \div \frac { a + 2 } { a ^ { 2 } - 2 a }$$

Evaluate

$$a$$

Short Solution Steps

Factor a^{2}-4.

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

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

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

Since \frac{\left(a-2\right)\left(a-2\right)}{\left(a-2\right)\left(a+2\right)} and \frac{8a}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a-2\right)\left(a-2\right)+8a.

$$\frac{\frac{a^{2}-2a-2a+4+8a}{\left(a-2\right)\left(a+2\right)}}{\frac{a+2}{a^{2}-2a}}$$

Combine like terms in a^{2}-2a-2a+4+8a.

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

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

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

Cancel out a+2 in both numerator and denominator.

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

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

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

Cancel out a+2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a-2 in both numerator and denominator.

$$a$$

Differentiate w.r.t. a

$$1$$

Steps Using Definition of a Derivative

Factor a^{2}-4.

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

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

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

Since \frac{\left(a-2\right)\left(a-2\right)}{\left(a-2\right)\left(a+2\right)} and \frac{8a}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a-2\right)\left(a-2\right)+8a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{2}-2a-2a+4+8a}{\left(a-2\right)\left(a+2\right)}}{\frac{a+2}{a^{2}-2a}})$$

Combine like terms in a^{2}-2a-2a+4+8a.

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

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

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

Cancel out a+2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a+2}{a-2}}{\frac{a+2}{a^{2}-2a}})$$

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

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

Cancel out a+2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}-2a}{a-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a\left(a-2\right)}{a-2})$$

Cancel out a-2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a)$$

The derivative of ax^{n} is nax^{n-1}.

$$a^{1-1}$$

Subtract 1 from 1.

$$a^{0}$$

For any term t except 0, t^{0}=1.

$$1$$