Which of the following has the greatest value

Question

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?

Answer 1

Hint:

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

and similarly for the other powers of $2$

Answer 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.