Question

Why do you multiply by a power of 10 when writing a repeating decimal as a rational number

Answer 1

The idea is to shift the number enough so that you can cancel the repeating digits by subtraction.

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Example

0.181818_18 multiplied by 100 gives

18.181818_18

Subtracting the original number gives

18.000000_00 . . . . . . all the repeating digits have been eliminated

When you multiply by 100 and subtract 1 times the original, you have multiplied by 99. Thus the value of the original number is 18/99 or 2/11.