Does 0.99 have a power (with a positive integer exponent) that is less than 0.000001?

What about 0.999, does 0.999999 have one? If so, name one such power for each!
Since $0.99 \cdot 1.01 = (1 - 0.01)(1 + 0.01) = 1 - 0.0001 = 0.9999$, which is less than 1, therefore (similarly to what we saw in the solution of problem 10) every power of it is also less than 1, that is, multiplying any power of 0.99 by the same power of 1.01 will result in a number less than 1. Thus, if some power of 1.01 is greater than 1 million, the same power of 0.99 will be less than one millionth. Therefore, according to the solution of problem 10:

$0.99^{99999900} \quad < 0.000001$ and similarly

$0.99^{999999999900} < 0.000000000001$, and furthermore

$0.999 \cdot 1.001 = 0.999999 < 1$ and $0.999999 \cdot 1.000001 = 0.999999999999 < 1$ using these:

$$
\begin{array}{ll}
0.999^{999999000} & < 0.000001 \\
0.999^{999999999999900} & < 0.000000000001 \\
0.999999^{999999000000} & < 0.000001 \\
0.999999^{999999999999000000} & < 0.000000000001
\end{array}
$$