Match a transformed sum to a scaled reference expression (clarified brackets): Evaluate A = {[(3^(−2))^(−5)]^(1/5)} + {[(4^(−3))^(−6)]^(1/6)} − 1 and express A as K × {[(2^(−3))^(−4)]^(1/4)}. Find K.

Introduction / Context:
This item uses nested exponents with negative and fractional powers. By simplifying each bracket carefully, the left-hand side turns into a simple integer. The right-hand side is a separate expression; the task is to express the left as a constant multiple K of the right and identify K.

Given Data / Assumptions:

• A = {[(3^(−2))^(−5)]^(1/5)} + {[(4^(−3))^(−6)]^(1/6)} − 1.
• Reference R = {[(2^(−3))^(−4)]^(1/4)}.

Concept / Approach:
Use (a^m)^n = a^(mn) and a^(−k) = 1/a^k. Work inside-out, simplify each bracket to a simple power of the base, then add and compare to R to extract K = A / R.

Step-by-Step Solution:

Verification / Alternative check:
Compute numerically: A = 72, R = 8, ratio 72/8 = 9. Everything matches exactly.

Why Other Options Are Wrong:

• 8: That is R, not the multiplier.
• 27, 1/3: Incorrect constants that would not satisfy A = K × R.

Common Pitfalls:
Dropping outer exponents or forgetting to multiply exponents in the correct order; also, misapplying negative exponent rules.

Final Answer: