Why is 0.3 to power of 2 0.09 and not 0.9?

Question

Given the following expressions:

0.3² = 0.09
0.025² = 0.000625

I can head count 3*3 and 25*25, which helps me solve the above expressions. But how do I know how many 0s there should be before "9" or "625"?

It seems to me that I need to add them until I get the the number of decimal cases of the base times the exponent (1*2 and 3*2, respectively). Is that right? And what is the rationale behind that?

Answer 1

Since $0.3=3\times\frac1{10}$,$$0.3^2=\left(3\times\frac1{10}\right)^2=9\times\frac1{100}=0.09.$$And, since $0.025=25\times\frac1{1000}$,$$0.025^2=625\times\frac1{1000000}=0.000625.$$

Answer 2

You can write it like this: $$ 0.3^2 = (3 \cdot 10^{-1})^2 = 3^2 \cdot 10^{-1 \cdot 2} = 9 \cdot 10^{-2} = 0.09 $$