The goal here is to subtract one thing from another such that it cancels out the repeating part. To do that, you need to *only* have the repeating part to the right of the decimal, and you need that for two different numbers.
To get the first number, we simply multiply by whatever power of 10 we need to shift the decimal so it is right before the repeating part. In this example, we need to shift the decimal 1 place to the right, so we need to multiply by 10^(1).
Then we need a second number that also only has the repeating part to the right. Which means we need to shift the decimal to the right again. We can't do it by just 1, because we already did that, so 2 is next so we shift it two places by multiplying by 10^(2).
We then subtract, the repeating 0.888... cancels out and you're left with:
90x = 26
x = 26/90 = 13/45
You *can* choose different powers of ten, so long as they are different and greater than or equal to what you need to make the right side of the decimal have only the repeating part. For example, we could have chosen to multiply by 10^(3) for our second number:
1000x - 10x = 288.888... - 2.888...
990x = 286
x = 286/990 = 26/90 = 13/45
So it just reduces to the same thing anyway.