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Question

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m × 5n, where, m, n are non-negative integers.

(i) 38

(ii) 13125

(iii) 780

(iv) 14588625

(v) 12922×57 [NCERT]

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Solution

(i) The given number is 38.

Clearly, 8 = 23 is of the form 2m × 5n, where m = 3 and n = 0.

So, the given number has terminating decimal expansion.

38=3×5323×53=3×1252×53=375103=3751000=0.375

(ii) The given number is 13125.

Clearly, 125 = 53 is of the form 2m × 5n, where m = 0 and n = 3.

So, the given number has terminating decimal expansion.

13125=13×2323×53=13×82×53=104103=1041000=0.104

(iii) The given number is 780.

Clearly, 80 = 24 × 5 is of the form 2m × 5n, where m = 4 and n = 1.

So, the given number has terminating decimal expansion.

780=7×5324×5×53=7×1252×54=875104=87510000=0.0875

(iv) The given number is 14588625.

Clearly, 625 = 54 is of the form 2m × 5n, where m = 0 and n = 4.

So, the given number has terminating decimal expansion.

14588625=14588×2424×54=14588×162×54=233408104=23340810000=23.3408

(v) The given number is 12922×57.

Clearly, 22 × 57 is of the form 2m × 5n, where m = 2 and n = 7.

So, the given number has terminating decimal expansion.

12922×57=129×2522×57×25=129×322×57=4182107=418210000000=0.0004182

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