Question

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating decimal expansion:
$\frac{7^3}{2^3\times 5^2}$ and $\frac{3\times 7^3}{2^3\times 3}$

• A. $\frac{7^3}{2^3\times 5^2}$ - Terminating
  
  $\frac{3\times 7^3}{2^3\times 3}$ - Non terminating
• B. Both have terminating decimal expansions.
• C. $\frac{7^3}{2^3\times 5^2}$ - Non terminating
  
  $\frac{3\times 7^3}{2^3\times 3}$ - Terminating
• D. Both have non-terminating decimal expansions.

Answer 1

The correct option is B

Both have terminating decimal expansions.

If the denominator of a rational number is in the form of $2^n{.5}^m$, where m and n are non-negative integers, then the rational number has terminating decimal expansion.

$\therefore$ $\frac{7^3}{2^3\times 5^2}$ has terminating decimal expansion
and
$\frac{3\times 7^3}{2^3\times 3}=\frac{7^3}{2^3\times 5^0}$ has terminating decimal expansion.