Question

Without actually performing the long division, state whether 105/
11200

will

have a terminating decimal expansion or a non-terminating repeating

decimal expansion.​

Answer 1

$\huge\purple{ANSWER:}$

AS FOR A DECIMAL EXPANSION OF A RATIONAL NUMBER TO BE TERMINATING, ITS DENOMINATOR SHOULD HAVE POWERS OF 2, 5 OR BOTH.

IN 105/11200 , THE DENOMINATOR HAS POWERS OF BOTH 2 AND 5, SO IT WOULD BE A TERMINATING DECIMAL.

Answer 2

Answer:

$\frac{105}{11200}$ is non-terminating decimal expansion.

Step-by-step explanation:

To Find:- Whether $\frac{105}{11200}$ a terminating decimal expansion or a non-terminating decimal expansion.

Solution:-

Let x= $\frac{a}{b}$ be a rational number, such that the prime factorization of b is of the form $2^{n} 5^{m}$, where n, m are non-negative integers.

Then, x has a decimal expansion which terminates.

Now, Factorize the denominator of $\frac{105}{11200}$ , we get

11200 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 7

          = $2^{6} 5^{2} 7^{}$

∵ Denominator in not in the form $2^{n} 5^{m}$.

∴ $\frac{105}{11200}$ is non-terminating decimal expansion.

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