Algebraic division with indices: Find the quotient of (a^(−1) − 1) divided by (a − 1).

Introduction / Context:
This checks algebraic manipulation with negative exponents and simple factorization. Rewriting a^(−1) as 1/a keeps the work transparent.

Given Data / Assumptions:

• Compute ((a^(−1) − 1) / (a − 1)) for a ≠ 0 and a ≠ 1.

Concept / Approach:
Rewrite a^(−1) as 1/a, factor out signs to match a − 1, and reduce the fraction carefully to avoid sign errors.

Step-by-Step Solution:
(a^(−1) − 1)/(a − 1) = ((1/a) − 1)/(a − 1)= ((1 − a)/a)/(a − 1)= ((−(a − 1))/a)/(a − 1)= (−1/a) * ((a − 1)/(a − 1)) = −1/a

Verification / Alternative check:
Pick a = 2: numerator = 1/2 − 1 = −1/2; denominator = 1; quotient = −1/2 = −1/a.

Why Other Options Are Wrong:
They come from sign mistakes or inverting the wrong term during simplification.

Common Pitfalls:
Dropping the minus sign when converting (1 − a) to −(a − 1); ensure domain excludes a = 0, 1.

Final Answer:
-1/a