Question

5/128 ,State whether given rational number have terminating decimal expansion or not and if it has terminating decimal expansion, find it.

Answer 1

If the factors of denominator of the given rational number is of form 2ⁿ 5^m ,where n and m are non negative integers, then the decimal expansion of the rational number is terminating otherwise non terminating recurring.

SOLUTION:

5/128 = 5 / 2^7

Here, the factors of the denominator 128 are are 5^0 × 2^7 which is in the form 2ⁿ 5^m .

5/128 has terminating decimal expansion which can be expressed as :

5/128 =5/128

= 5 × 5^7 / 5^7× 2^7

= 390625/(5×2)^7

= 390625/10^7

= 390625/10000000

= 0.0390625

Hence, the terminating decimal expansion of 5/128 is 0.0390625

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Answer 2

Hey there!

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Q. $\frac{5}{128}$

⇒ Given number is $\frac{5}{128}$ and HCF (5, 128) = 1

∴ $\frac{5}{128} = \frac{5 ÷ 1}{128 ÷ 1} = \frac{5}{128}$

⇒ Now, 128 = $(2^{7})$ and 2 is not the factor of 5.

∴ $\frac{5}{128}$ is in its simplest form.

⇒ Also, 128 = $(2^{7}) = (2^{m} × 5^{n})$

∴ $\frac{5}{128}$ is a terminating decimal.

⇒ $\frac{5}{128}$ has a terminating decimal expansion which can be expressed as

∴ $\frac{5}{128} = \frac{5 × 5^{7}}{2^{7} × 5^{7}} = \frac{390625}{10000000} = 0.0390625$