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Number of digits

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Number of digits  [#permalink]

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New post 03 Jun 2015, 21:25
Number of digits in 2 to power 2 to power 22
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New post 03 Jun 2015, 21:26
powers are not in multiplication
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New post 03 Jun 2015, 23:55
Hi ayushthebest,

What is the source of this question? I ask because if this were a GMAT Problem Solving question, then there would be 5 answer choices to work with (and it would likely be that the answers would be written in such a way as to provide you with a way to avoid doing lots of work or at least estimate a reasonable guess).

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New post 04 Jun 2015, 11:38
2
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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New post 29 Jan 2018, 00:11
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Re: Number of digits &nbs [#permalink] 29 Jan 2018, 00:11
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