Question

The decimal expansion of the rational number $\dfrac{43}{{{3}^{4}}\times {{5}^{2}}}$ will terminate after how many decimals?

Answer 1

Hint: In this question, we are given prime factorization of the denominator of a number. We have to tell after how many numbers will the decimal expansion of this number terminate. For this, we will use the concept of terminating decimals which states that a rational number terminates if its denominator has factors of 2 and 5 only otherwise the number never terminates but repeats in decimal expansion.

Complete step by step solution:
Here, we are given numbers as $\dfrac{43}{{{3}^{4}}\times {{5}^{2}}}$. Now, the denominator of this number is expressed in prime factors and we have to evaluate when will the number terminate in its decimal expansion. For this, let us understand the concept of terminating decimals. If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expansion terminates.
If there is a prime factor in the denominator other than 2 or 5, then the decimal expansion repeats. Now, let us observe the denominator of the given number. It has prime factors of 3 and 5. Since it has prime factors other than 2 or 5, so this decimal expansion will repeat. Hence, the decimal expansion of $\dfrac{43}{{{3}^{4}}\times {{5}^{2}}}$ will never terminate.

Note: Students should note that any terminating decimal is essentially a fraction with a power of ten in the denominator. For example, $0.0376=\dfrac{376}{10000}=\dfrac{47}{1250}$ which may get simplified later but denominators can only have factors 2 or 5 for decimal expansion to terminate. If the denominator of number has prime factors of the form ${{2}^{m}}\times {{5}^{n}}$ then the numbers terminate after m places, if m>n and it terminates after n places, if n>m.