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Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) ` (13)/(3125) `
(ii) ` (17)/8 `
(iii) ` (64)/(455) `
(iv) ` (15)/1600 `
(v) ` (29)/(343) `
(vi) ` (23)/((2^3).(5^2)) `
(vii) ` (129)/((2^2)(5^7)(7^5)) `
(viii) ` (6)/(15) `
(ix) ` (35)/(50) `
(x) ` (77)/(210) `

Text Solution

AI Generated Solution

To determine whether the given rational numbers have a terminating or non-terminating repeating decimal expansion, we need to analyze the prime factorization of their denominators. A rational number will have a terminating decimal expansion if the prime factorization of its denominator (after simplifying the fraction) consists only of the primes 2 and 5. If there are any other prime factors, the decimal expansion will be non-terminating repeating. Let's analyze each of the given rational numbers step by step: ### Step 1: Analyze each rational number **(i) \( \frac{13}{3125} \)** - Prime factorization of 3125: ...
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