Problem 77 - College Algebra Essentials

Question

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+5)^{-1}}{5}$$

Step-by-Step Solution

Answer

\(\frac{1}{y*(y+5)}\)

Step 1: Simplify the numerator

To simplify the numerator, first rewrite \(y^{-1}\) and \((y+5)^{-1}\) as \(\frac{1}{y}\) and \(\frac{1}{y+5}\) respectively. Then, find a common denominator to subtract the fractions, which is \(y*(y+5)\). So the new numerator becomes: \(\frac{y+5-y}{y*(y+5)} = \frac{5}{y*(y+5)}\).

Step 2: Simplify the fraction by division

The original fraction is now \(\frac{\frac{5}{y*(y+5)}}{5}\). We simplify this by dividing 5 out from the numerator and denominator which results in \(\frac{1}{y*(y+5)}\).

Step 3: Simplify further

The division can be expressed as multiplication by the reciprocal. So the final answer is \(\frac{1}{y*(y+5)}\).