What type of decimal expansion will (\frac{2^5\cdot 7}{2^8\cdot 5^2\cdot 7^2}) have?
Answer and explanation
Correct answer: Non-terminating recurring
For a rational number, the decimal expansion terminates only when, in lowest terms, the denominator contains powers of 2 and 5 alone. Any remaining prime factor other than 2 or 5 makes the decimal expansion non-terminating recurring. Thus the important step is to cancel common factors completely before applying this rule.
Cancel the common factors in \(\frac{2^5\cdot7}{2^8\cdot5^2\cdot7^2}\). The result is \(\frac{1}{2^3\cdot5^2\cdot7}\), because \(2^5\) leaves \(2^3\) below and one factor 7 remains below. Since 7 is still a factor of the reduced denominator, the decimal is non-terminating recurring. Therefore option B is correct. It cannot terminate after three places, because the factor 7 prevents termination.
Frequently asked questions
What is the correct answer to this question?
Non-terminating recurring
Why is this the correct answer?
For a rational number, the decimal expansion terminates only when, in lowest terms, the denominator contains powers of 2 and 5 alone. Any remaining prime factor other than 2 or 5 makes the decimal expansion non-terminating recurring. Thus the important step is to cancel common factors completely before applying this rule.
Cancel the common factors in \(\frac{2^5\cdot7}{2^8\cdot5^2\cdot7^2}\). The result is \(\frac{1}{2^3\cdot5^2\cdot7}\), because \(2^5\) leaves \(2^3\) below and one factor 7 remains below. Since 7 is still a factor of the reduced denominator, the decimal is non-terminating recurring. Therefore option B is correct. It cannot terminate after three places, because the factor 7 prevents termination.
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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