The bracketing matters a lot here. If the question is asking about
\((2^2)^{22}\)
then we can use the fact that 2^10 is just slightly bigger than 1000, so is just slightly bigger than 10^3. So since
\((2^2)^{22} = 2^{44} = (2^4)(2^{40}) = (2^4)(2^{10})^4 \approx 16 \times (10^3)^4 = 16 \times 10^{12}\)
then since 10^12 is a one followed by twelve zeros, then our estimate will be a 16 followed by twelve zeros, so will have fourteen digits. Our estimate is close enough that it will have the same number of digits as 2^44.
But if instead the question is asking about:
\(2^{2^{22}}\)
I don't see an easy way to get an exact answer. 2^22 = (2^2) (2^20) = (2^2) (2^10)^2, which as we saw above is roughly (4)(10^3)^2 = 4,000,000. So we want to know how many digits are in a number a bit bigger than 2^(4,000,000). Again using the same estimate, 2^(4,000,000) = (2^10)^(400,000) ~ (10^3)^(400,000) = 10^(1,200,000). So the right answer would be bigger than 1.2 million, but our estimate is no longer very precise here, since we're raising things to such huge powers.
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