Numerical

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^{m} × 5^{n}, where, *m*, *n* are non-negative integers.\[\frac{13}{125}\]

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#### Solution

The given number is \[\frac{13}{125}\].

Clearly, 125 = 5^{3} is of the form 2^{m} × 5^{n}, where *m* = 0 and *n* = 3.

So, the given number has terminating decimal expansion.

\[\therefore \frac{13}{125} = \frac{13 \times 2^3}{2^3 \times 5^3} = \frac{13 \times 8}{\left( 2 \times 5 \right)^3} = \frac{104}{\left( 10 \right)^3} = \frac{104}{1000} = 0 . 104\]

Concept: Fundamental Theorem of Arithmetic

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