Question

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m × 5ⁿ, where m, n are non-negative integers.

\[\frac{129}{2^2 \times 5^7}\]

Answer 1

The given number is \[\frac{129}{2^2 \times 5^7}\]. 

Clearly, 2² × 5⁷ is of the form 2^m × 5ⁿ, where m = 2 and n = 7.

So, the given number has terminating decimal expansion.

\[\therefore \frac{129}{2^2 \times 5^7} = \frac{129 \times 2^5}{2^2 \times 5^7 \times 2^5} = \frac{129 \times 32}{\left( 2 \times 5 \right)^7} = \frac{4182}{\left( 10 \right)^7} = \frac{4182}{10000000} = 0 . 0004182\]