Difficulty: Easy
Correct Answer: 4
Explanation:
Introduction / Context:Recognizing prime-power structure in decimals allows fast simplification. Here, 0.49, 0.343, and 0.2401 are powers of 7 divided by matching powers of 10. The expression reduces neatly to a single power of 7/10.
Given Data / Assumptions:
Concept / Approach:Rewrite each decimal: 0.49 = 7^2/10^2, 0.343 = 7^3/10^3, 0.2401 = 7^4/10^4. Apply exponent rules to combine and subtract powers when dividing.
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.E = (7^(8+12)/10^(8+12)) / (7^16/10^16) = 7^(20−16)/10^(20−16) = (7/10)^4.Therefore k = 4.Verification / Alternative check:Since 0.7^4 = 0.2401, the conclusion that E = (0.7)^4 is consistent with the given numbers.
Why Other Options Are Wrong:1, 2, 3, and 7 do not match the net exponent after combining powers; only 4 preserves the equality.
Common Pitfalls:Adding instead of subtracting exponents during division, or misidentifying the decimal-to-fraction conversions.
Final Answer:4
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