Question

The decimal expansion of the rational no. $ \dfrac{{43}}{{{2^4}{5^3}}} $ will terminate after how many decimals?

Answer 1

Hint: A rational number is the number which can be expressed as the ratio of two numbers or which can be expressed as the p/q form or as the quotient or the fraction with non-zero denominator. Here we will simplify the given fraction and will find the equivalent fraction and then will find the required number.

Complete step-by-step answer:
Take the given expression –
 $ \dfrac{{43}}{{{2^4}{5^3}}} $
It can be re-written as –
 $ = \dfrac{{43}}{{{2^4} \times {5^3}}} $
Multiply and divide the expression with . when you multiply and divide the term with the same value, the ultimate value remains the same.
 $ = \dfrac{{43 \times 5}}{{{2^4} \times {5^3} \times 5}} $
By using the law of power and exponent, when there is a common base and multiplicative sign in between then powers are added.
 $ = \dfrac{{43 \times 5}}{{{2^4} \times {5^{3 + 1}}}} $
Simplify the above expression –
 $ = \dfrac{{43 \times 5}}{{{2^4} \times {5^4}}} $
Again using the law of power and exponent – when powers are equal, then base are multiplied.
 $ = \dfrac{{43 \times 5}}{{{{\left( {2 \times 5} \right)}^4}}} $
Simplify the above expression –
 $ = \dfrac{{43 \times 5}}{{{{\left( {10} \right)}^4}}} $
Simplify the above expression –
 $ = \dfrac{{215}}{{10000}} $
Convert the above fraction in decimal point.
 $ = 0.0215 $
Hence, the given rational number will terminate after four decimal places.
So, the correct answer is “4”.

Note: The numbers which are not represented as the rational are known as the irrational number. Remember zero is the rational number. Also, refer to other terminologies for natural numbers, whole numbers and integers, fractions and know the difference between them. Always remember that when you multiply and divide the term with the same number the resultant value remains the same. Be careful while converting the fraction into decimal points. Always shift the points from the right hand side of the number.