What type of decimal expansion will 14/(2^2 × 5^3 × 7^2) have?
Answer and explanation
Correct answer: Non-terminating recurring
The denominator must be examined after cancelling common factors. Factor the numerator as 14 = 2 × 7. Cancelling these factors from 2^2 × 5^3 × 7^2 leaves the reduced denominator 2 × 5^3 × 7. The criterion says that a rational number has a terminating decimal only when its reduced denominator has no prime factors other than 2 and 5. The remaining factor 7 prevents termination. Because the number is rational, its non-terminating decimal is recurring rather than non-recurring. Therefore option B is correct. Option A is wrong because the factor 7 remains; option D is also wrong for the same reason, and option C would describe a non-rational decimal rather than this rational number.
Frequently asked questions
What is the correct answer to this question?
Non-terminating recurring
Why is this the correct answer?
The denominator must be examined after cancelling common factors. Factor the numerator as 14 = 2 × 7. Cancelling these factors from 2^2 × 5^3 × 7^2 leaves the reduced denominator 2 × 5^3 × 7. The criterion says that a rational number has a terminating decimal only when its reduced denominator has no prime factors other than 2 and 5. The remaining factor 7 prevents termination. Because the number is rational, its non-terminating decimal is recurring rather than non-recurring. Therefore option B is correct. Option A is wrong because the factor 7 remains; option D is also wrong for the same reason, and option C would describe a non-rational decimal rather than this rational number.
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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