Problem 78 - College Algebra

Question

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+2)^{-1}}{2}$$

Step-by-Step Solution

Answer

\( \frac{1}{y(y+2)} \)

Step 1: Transform Negative Exponent to Positive

Transform the negative exponents into positive ones by moving the term from the numerator to the denominator or vice versa. This leads to the following expression: \[ \frac{1/y - 1/(y+2)}{2}\]

Step 2: Find Common Denominator

Find a common denominator for the terms in the numerator. We multiply \(1/y\) by \((y+2)/(y+2)\) and \(1/(y+2)\) by \(y/y\) to have the same denominator. This results in \[ \frac{(y+2) -y }{2y(y+2)}\]

Step 3: Simplify the Result

Simplify the resulting fractions. The numerator becomes \(2\). The final answer is \[ \frac{2}{2y(y+2)} = \frac{1}{y(y+2)}\].