# Which of the following has the greatest value

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?

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

• $$16^{17}=\left(2^4\right)^{17}=2^{4\times17}=2^{68}$$
and similarly for the other powers of $$2$$
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.