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You are correct. Nowhere does the number 3 appear in the multiplication. The number of factors involved is different from the factors themselves. We can use one factor any number of times.

Now, if we had to write out the multiplication in which 2 is used as a factor nine times, it would be rather cumbersome:

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512.

For convenience, there is a shorthand way of writing a phrase such a “the product formed by using 2 as a factor nine times.” The whole phrase may be written simply as “2^{9}.” This symbol means

thenumberformedbyusingthisnumberasafactor → 2^{9 ← this many times}