If the reduced denominator is q = 2^5 × 5^5 × 7^0, what is certain about the decimal expansion?
Answer and explanation
Correct answer: Terminates exactly after 5 places
Because 7^0 = 1, the denominator is effectively 2^5 × 5^5. These powers combine to give 10^5, since 2^5 × 5^5 = (2 × 5)^5 = 10^5. Thus the fraction has a denominator of 100000 after reduction and its decimal expansion terminates within five places. Moreover, because the fraction is in lowest form, the numerator is coprime to both 2 and 5. It therefore cannot supply a factor that would cancel the final power of 10; the fifth decimal digit cannot become an unnecessary trailing zero. Hence the expansion terminates exactly after five places, making option A correct. Option B incorrectly adds exponents, while options C and D contradict the fact that the denominator contains only 2 and 5.
Frequently asked questions
What is the correct answer to this question?
Terminates exactly after 5 places
Why is this the correct answer?
Because 7^0 = 1, the denominator is effectively 2^5 × 5^5. These powers combine to give 10^5, since 2^5 × 5^5 = (2 × 5)^5 = 10^5. Thus the fraction has a denominator of 100000 after reduction and its decimal expansion terminates within five places. Moreover, because the fraction is in lowest form, the numerator is coprime to both 2 and 5. It therefore cannot supply a factor that would cancel the final power of 10; the fifth decimal digit cannot become an unnecessary trailing zero. Hence the expansion terminates exactly after five places, making option A correct. Option B incorrectly adds exponents, while options C and D contradict the fact that the denominator contains only 2 and 5.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
