Number of digits

Question

Number of digits in 2 to power 2 to power 22

ayushthebest · Answer

powers are not in multiplication

EMPOWERgmatRichC · Answer

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).

GMAT assassins aren't born, they're made,
Rich

IanStewart · Answer

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 	imes (10^3)^4 = 16 	imes 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.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Number of digits

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards