The sum of all real values of k for which (18)^k × (1/32768)^(1/3) = 18 × (1/32768)^(1/k), is

Question

UPSC Quizroom · Accepted Answer

Equation: (1/8)^k × (1/32768)^(1/3) = (1/8) × (1/32768)^(1/k)

Step 1: Express as powers of 2:

1/8 = 2⁻³, 1/32768 = 2⁻¹⁵

→ 2^(−3k) × 2^(−15/3) = 2^(−3) × 2^(−15/k)
→ 2^(−3k − 5) = 2^(−3 − 15/k)

Step 2: Equate exponents:

−3k − 5 = −3 − 15/k → 3k² + 2k − 15 = 0

Step 3: Sum of roots of quadratic = −b/a = −2/3

🎯 UPSC Quiz Room