What type of decimal expansion will 22/(2^2 × 5^4 × 11^2) have?
Answer and explanation
Correct answer: Non-terminating recurring
Reduce the fraction first. The numerator is 22 = 2 × 11. Cancelling these factors from the denominator 2^2 × 5^4 × 11^2 leaves 2 × 5^4 × 11. A rational number has a terminating decimal only if every prime factor in its reduced denominator is 2 or 5. Although the factors 2 and 5 would support termination, the factor 11 remains after cancellation, so the denominator is not a power of 10 and the decimal cannot terminate. The number is rational, therefore its infinite decimal expansion is eventually periodic, or recurring. Hence option B is correct. Option A and option D incorrectly assume that the factors 2 and 5 alone decide the result before reduction; option C incorrectly treats a rational decimal as non-recurring.
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What is the correct answer to this question?
Non-terminating recurring
Why is this the correct answer?
Reduce the fraction first. The numerator is 22 = 2 × 11. Cancelling these factors from the denominator 2^2 × 5^4 × 11^2 leaves 2 × 5^4 × 11. A rational number has a terminating decimal only if every prime factor in its reduced denominator is 2 or 5. Although the factors 2 and 5 would support termination, the factor 11 remains after cancellation, so the denominator is not a power of 10 and the decimal cannot terminate. The number is rational, therefore its infinite decimal expansion is eventually periodic, or recurring. Hence option B is correct. Option A and option D incorrectly assume that the factors 2 and 5 alone decide the result before reduction; option C incorrectly treats a rational decimal as non-recurring.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.
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