Problem 77 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Operations

The task is to perform the operations requested in this expression: \( \frac{y^{-1}-(y+5)^{-1}}{5} \). Here, we'll deal with exponents and division.

Step 2: Convert Negative Exponents to Positive

A negative exponent means we need to take the reciprocal of the base. Hence \(y^{-1} = \frac{1}{y}\) and \((y+5)^{-1} = \frac{1}{y+5}\). Our expression becomes: \( \frac{\frac{1}{y} - \frac{1}{y+5}}{5} \)

Step 3: Simplify the Numerator

To be able to subtract fractions, they need to have a common denominator. Hence we will multiply them by the other fraction's denominator, giving us: \( \frac{\frac{1}{y}*(y+5) - \frac{1}{y+5}*y}{5} \). Simplifying the numerator further gives us \( \frac{y+5 - y}{5(y+5)} \)

Step 4: Simplify Further

The y in the numerator cancels out, leaving us with \( \frac{5}{5(y+5)} \). Division can also be seen as multiplication by the reciprocal, so we can simplify this to: \( \frac{1}{y+5} \)

Step 5: Final Answer

The simplified result of the expression \( \frac{y^{-1}-(y+5)^{-1}}{5} \) is \( \frac{1}{y+5} \)