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:
Note: - If x be a rational number, then x can be expressed in the form p/q where p and q are Co- primes. Then, if the prime factorization of denominator (q) is in the form of 2m × 5n where n, m are non- negative integers, then x has a decimal expansion which terminates. If the prime factorization is not in this form of 2m × 5n then x has a decimal expansion which is non terminating.
(i) 13 / 3125
= 5 × 5 × 5 × 5 × 5 = 5⁵ (on factorization of q)
Since, its factorization contains only power of 5.
Therefore, it has a terminating decimal expansion.
(ii) 17 / 5
= 2 × 2 × 2 = 2³ (on factorization of q)
Since, its factorization contains only power of 2.
Therefore, it has a terminating decimal expansion.
(iii) 64 / 455
= 5 × 7 × 13 = 5¹ × 7 × 13 (on factorization of q)
Since the factorization of not in the 2m ×5n
Therefore, it has a non -terminating decimal expansion.
(iv) 15 / 1600
= 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 2⁶ × 5² (on factorization of q)
Since the factorization of q is in the form 2m ×5n
Therefore, it has a terminating decimal expansion
(V) 29 / 343
= 7 × 7 × 7 = 7³ (on factorization of q)
Since the factorization of denominator is not in the form 2m × 5n
Therefore it has a non -terminating repeating decimal
(vi) 23 / 23 52
Since factorization of q already given and it is in the form 2m × 5n
Therefore it has a terminating decimal expansion.
(vii) 129 / 22 57 75
Since factorization of q already given but it is not in the form 2m × 5n
Therefore it has a terminating decimal expansion.
(viii) 6 / 15 = 2/5
5 = 5 × 1 (on factorization of q)
Since 5 is the only factor in denominator.
Therefore it has a non -terminating repeating decimal expansion.
(ix). 35 / 50 = 7 / 10
10 = 2 × 5 = 2¹ × 5 (on factorization of q)
Since the factorization of denominator is in the form 2m × 5n
Therefore it has a terminating decimal expansion.
(x) 77 / 210 = 11/30
30 = 2 × 3 × 5 × 7 = 2¹ × 3 × 5 × 7. (on factorization of q)
Since the factorization of denominator is not in the form 2m × 5n
Therefore it has a non -terminating repeating decimal expansion.
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Welcome to the NCERT Solutions for Class 10 Mathematics - Chapter . This page offers a step-by-step solution to the specific question from Excercise 4 , Question 1: Without actually performing the long division, state whether the following rational numbers will hav....
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