Problem 78
Question
Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+2)^{-1}}{2}$$
Step-by-Step Solution
Verified Answer
The simplified form of the given expression is \(-\frac{1}{y(y+2)}\).
1Step 1: Consolidate the terms in the numerator
First, we need to consolidate the terms in the numerator. As subtraction is involved, first find a common denominator for the two fractions, which will be \( y \cdot (y+2) \). So the fraction transforms to: \[ \frac{y - (y+2)}{2y(y+2)} \]
2Step 2: Simplify the numerator
Simplify the numerator by performing the subtraction. This gives: \[\frac{y - y - 2}{2y(y+2)}\] Simplify to: \[\frac{- 2}{2y(y+2)}\]
3Step 3: Simplify the fraction
Continue simplifying the fraction by canceling out the common factors in the numerator and denominator. This will give: \[\frac{-1}{y(y+2)}\]
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