Question

Let x = \(\frac{7}{2^2\ \times \ 5^3}\) be a rational noumber. Then x has a decimal expansion which
terminates 
1. After 4 places of decimal 
2. After 3 places of decimal 
3. After 2 places of decimal
4. After 5 places of decimal

Answer 1

Correct Answer - Option 2 : After 3 places of decimal 

Concept:

Rational Numbers:

1. A number of the form p/q, where p and q are integers and q ≠ 0 are called rational numbers.

2. Square roots of prime numbers are irrational.

Formula used:

a^mb^m = (ab)^m

Calculation:

Given that,

x = \(\frac{7}{2^2\ \times \ 5^3}\)

⇒ x = \(\frac{7\times2}{2^3\ \times \ 5^3}\)

∵ ambm = (ab)m

⇒ x = \(\frac{14}{10^3}\ =\ \frac{14}{1000}\)

⇒ x = 0.014

Hence, x will terminate after 3 places of decimal. 

Theorem:  Let x be a rational number whose simplest form is p/q, where

p and q are integers and q ≠ 0. Then, x is a terminating decimal only when q is of form 2^m x 5ⁿ for some non-negative integers m and n.