Question

the decimal expansion of a rational number 9/(2×5⁴) will terminate after​

Answer 1

Step-by-step explanation:

2

2

⋅5

33

=

4×5×5

33×5

=

100

165

=1.65.

Thus the decimal expansion will terminate after two decimal places.

Therefore, option B is correct.

Answer 2

The decimal expansion of the given rational number will terminate after 4 decimal places.

Rational numbers are the numbers that can be expressed in fractional form that is of the form $\frac{p}{q}$ where the denominator q is a non-zero value [$q\neq 0$] as it will then become an undefined value.

The p and q values are integer values which may include both negative as well as positive whole numbers.

Since, a rational is a fraction it can also be expressed as decimals. Thus, all decimal numbers whether it is recurring or non-recurring are also known as rational numbers.

An un-simplified version of a rational number is given and the decimal expansion of the same is to be found.

The fraction is given as: $\frac{9}{2\times 5^{4} }$

The fraction first needs to brought to its simplest form as the denominator must be solved.

On further simplification we get:

$\implies \frac{9}{2\times 625}$   [∵$5^{4}=25\times 25=625$]

$\implies \frac{9}{1250}$

This is the simplest form of the rational number.

Now, converting the above fraction into decimal form by dividing the numerator by the denominator. The resulting decimal expansion is given as:

$\frac{9}{1250}=0.0072$

It is asked when the decimal expansion will terminate. This means that after how many decimal places will the decimal point reach a position where the decimal number becomes into a whole number.

Thus, when we shift the decimal places by 4 places towards the right the decimal point will be positioned after 2 and the decimal number becomes a whole number 72.

Hence, the decimal expansion of a rational number $\frac{9}{2\times 5^{4}}$ will terminate after 4 decimal places.

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