Question

The decimal expansion of the rational number $\dfrac{{23457}}{{{2^3} \times {5^4}}}$ will terminate after how many places of decimals?
A. 2
B. 3
C. 4
D. 5

Answer 1

Hint: Try forming multiples of 10 in the denominator as it is easy to divide the number by 10. These multiples can be formed by taking out factors which are 2 and 5 such that the individual power of factor 2 and factor 5 comes equal.

Complete step-by-step answer:
We are given a rational number $\dfrac{{23457}}{{{2^3} \times {5^4}}}$ . And we need to calculate after how many decimals after expanding it in decimal form.
First, we will start by understanding what the question is asking for. The question is after how many places of decimals will the decimal form of the given rational number be terminated.
It means to say that when the rational number is divided by the denominator how many decimals are there in it.
One way is to calculate the entire denominator by multiplying all the numbers and then divide the numerator. This process is lengthy and chances of mistakes are also high.
This solution can be simplified by forming factors of 10 in the denominator.
10 can be formed by multiplying 2 and 5. Therefore, we need to take out factors 2 and 5 such that the powers of them are equal. This can be shown as below –
\[\dfrac{{23457}}{{{2^3} \times {5^4}}} = \dfrac{{23457}}{{{2^3} \times {5^3} \times 5}}\]
\[ \Rightarrow \dfrac{{23457}}{{{{10}^3} \times 5}}\] ……………. Using the property \[{a^3} \times {b^3} = {(a.b)^3}\]
Now, to make a factor of 10 again in the denominator multiplying 2 in both the denominator and numerator. Thus,
\[ \Rightarrow \dfrac{{23457 \times 2}}{{{{10}^3} \times 5 \times 2}} = \dfrac{{23457 \times 2}}{{{{10}^3} \times 10}}\]
Simplifying the numerator and denominator we get,
\[ \Rightarrow \dfrac{{46914}}{{{{10}^4}}}\]
Dividing by \[{10^4}\] we get,
 \[ \Rightarrow \dfrac{{46914}}{{{{10}^4}}} = 4.6914\]
After 4 decimals the decimal expansion is terminated.

Therefore, option (c) is the correct answer.

Note: Only a number which is rational can be terminated if the number given is irrational then the decimals are recurring in nature therefore never terminated.