Simplify (a^-1+4)/(a^-1-4)

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

Step 1

Simplify the numerator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

To write $4$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{\frac{1 + 4 a}{a}}{\frac{1}{a} - 4 \cdot \frac{a}{a}}$

Combine $- 4$ and $\frac{a}{a}$ .

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

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 + 4 a}{a} \cdot \frac{a}{1 - 4 a}$

Step 4

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{1 + 4 a}{a} \cdot \frac{a}{1 - 4 a}$

Rewrite the expression.

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

Step 5

Multiply $1 + 4 a$ by $\frac{1}{1 - 4 a}$ .

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