Question

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

Answer 1

Suppose you have some repeating decimal like

x = 0.123123123...

The repeating part has a length of 3, and multiplying both sides by 10^3 = 1000 gives

1000x = 123.123123123...

The usefulness of this is that we can subtract x from this to eliminate the decimal part:

1000x - x = 123.123... - 0.123...

999x = 123

then solve for x as a fraction:

x = 123/999

and in this case the fraction reduces further to

x = 41/333