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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

(i) \(\frac{23}{8}\)

(ii) \(\frac{125}{441}\)

(iii) \(\frac{35}{50}\)

(iv) \(\frac{77}{210}\)

(v) \(\frac{129}{2^2\times 5^7\times 7^{17}}\)

1 Answer

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Best answer

(i) 8 = 23 since the denominator is of the form 2m therefore the expression \(\frac{23}{8}\) is terminating.

(ii) 441 = 32 × 72 since the denominator is not in the form of 2m and 5n

Therefore the given expression is Non - terminating repeating

(iii) \(\frac{7\times5}{10\times5}\) = \(\frac{7}{10}\)

10 = 2×5 therefore the denominator is of the form 2m × 5n

Hence, the given expression is Terminating

(iv) \(\frac{77}{210}\)\(\frac{7\times11}{30\times7}\)\(\frac{11}{30}\)

30 = 2×3×5

Since the denominator is not of the form 2m × 5n

Hence, the given expression is Non - terminating repeating Terminating

(v) \(\frac{129}{2^2\times 5^7\times 7^{17}}\)

Since the denominator is not in the form 2m × 5n

Hence, the given expression is Non - terminating repeating Terminating

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