Difficulty: Easy
Correct Answer: 4
Explanation:
Introduction / Context:This problem checks your ability to convert recurring decimal bases into powers of a common prime (here, 7) over powers of 10 and then simplify using exponent rules. The goal is to match the simplified result to a clean form (70/100)^?.
Given Data / Assumptions:
Concept / Approach:Rewrite each decimal with base 7 and 10: 0.49 = 7^2/10^2, 0.343 = 7^3/10^3, 0.2401 = 7^4/10^4. Apply exponent laws to combine the powers, then simplify the net exponents to match the target base 70/100 = 7/10.
Step-by-Step Solution:
(0.49)^4 = (7^2/10^2)^4 = 7^8/10^8.(0.343)^4 = (7^3/10^3)^4 = 7^12/10^12.(0.2401)^4 = (7^4/10^4)^4 = 7^16/10^16.Combine: (7^8·7^12)/(7^16) over (10^8·10^12)/(10^16) = 7^(20−16) / 10^(20−16) = 7^4/10^4.Since 7^4/10^4 = (7/10)^4 = (70/100)^4, the required exponent is 4.Verification / Alternative check:Numerically, 0.7^4 = 0.2401, which is consistent with the original decimal pattern.
Why Other Options Are Wrong:3, 1, 2, and 7 do not match the simplified exponent balance. Only 4 preserves the equality.
Common Pitfalls:Mixing addition and subtraction of exponents, or failing to recognize that dividing by (7^4/10^4)^4 subtracts powers in both numerator and denominator.
Final Answer:4
Discussion & Comments