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Which of the following has the greatest value?

a) $2^{64}$ b) $4^{63}$ c) $8^{34}$ d) $16^{17}$

I tried finding a pattern among exponents and their is none. but there is a pattern in base, but I'm unable to find the common power through which I'll compare the base and figure the answer. What is the best possible option to solve this question within 1.5 minutes?

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    $\begingroup$ Express all of them as exponents of two $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Nov 30 '18 at 10:46
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    $\begingroup$ There's a very clear pattern in the bases: $4 = 2^2$, for a start. $\endgroup$ – user3482749 Nov 30 '18 at 10:46
  • $\begingroup$ Please read this tutorial on how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig Nov 30 '18 at 12:41
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Hint:

  • $16^{17}=\left(2^4\right)^{17}=2^{4\times17}=2^{68}$

and similarly for the other powers of $2$

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  • $\begingroup$ Ohh I got it, It was so right in the front, I was being stupid. Thank you so much. Correct answer is B right? $\endgroup$ – shawn k Nov 30 '18 at 10:52
  • $\begingroup$ Can you please help me with this question as well? I'm stuck and not getting any replies math.stackexchange.com/questions/3019103/… $\endgroup$ – shawn k Nov 30 '18 at 10:56
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    $\begingroup$ @shawn k I’ve answered your question. $\endgroup$ – KM101 Nov 30 '18 at 11:37
  • $\begingroup$ @KM101 Thanks a-lot!!! Really appreciated. $\endgroup$ – shawn k Nov 30 '18 at 11:44
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We know that $$16^{17}=4^{34}<4^{63}\\2^{64}=4^{32}<4^{63}\\8^{34}=4^{34\times {3\over 2}}=4^{51}<4^{63}$$ therefore $4^{63}$ has the greatest value among all.

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