Number series – descending base with ascending exponent (k^(k+1)): 81, 512, 2401, 7776, 15625, 16384, ? Continue the pattern where the base decreases by 1 and the exponent increases by 1.

Introduction / Context:This elegant pattern builds perfect powers with a decreasing base and an increasing exponent, creating familiar landmarks: 3^4=81, 8^3=512, 7^4=2401, 6^5=7776, 5^6=15625, 4^7=16384. The natural continuation is 3^8 followed by 2^9, etc., depending on where you start reading pairs.

Given Data / Assumptions:

• Observed chain after 4^7=16384 is 3^8 then 2^9.
• Among the choices, only 512 equals 2^9.

Concept / Approach:Track (base, exponent) as (8,3), (7,4), (6,5), (5,6), (4,7), (3,8), (2,9). Select the next term compatible with that ordered pair.

Step-by-Step Solution:

Verification / Alternative check:The uniform (−1, +1) shift in (base, exponent) between successive terms confirms the rule.

Why Other Options Are Wrong:

Common Pitfalls:Assuming a single fixed base; here both base and exponent evolve.

Final Answer:512