Question

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

Answer 1

Answer:

We'd like to take a repeating decimal and express it as a quotient of two whole numbers. The idea is to multiply by some number (10, 100, 1000, etc.) so that when we subtract the original number from the multiple, the repeating part cancels out. If we multiply by 10, we get 10x = 2.

Answer 2

By mulitplying with a power of 10, it moves the repeating digits immediately to the left of the decimal point (in other words, to eliminate any zeros preceding the repeating digits).