Question

[A bit challenging problem] Which of the following is largest: 3^210, 7^140, 17^105, or 31^84? Explain your answer.

Answer 1

Solve it with Python?

Answer 2

This can be solved in a line of python!

Answer 3

Anyone not using Python?

Answer 4

You can (obviously) use a calculator or even the google search window to evaluate each expression. For example, type
210*log(3)=
in the google search window.

However, you do not need a calculator. Starting with
\[3^{210}7^{140} 17^{105} 31^{84}\]
we can approximate these expressions by replacing the base with the nearest power of 2:
3 -> 2^2 (4) , 7 -> 2^3 (8), 17-> 2^4 (16), 31-> 2^5 (32)
from which we get
\[2^{2\cdot 210}= 2^{420}\]
\[2^{3\cdot 140}= 2^{420}\]
\[2^{4\cdot 105}= 2^{420}\]
\[2^{5\cdot 84}= 2^{420}\]
So all of them are approximately equal. BUT notice that three of our approximations are larger than the original expression (for example, 3^210 < 2^420 ), but one of the original expression is greater than the approximation : 17^105 > 16^105
Conclusion: 17^105 is larger than the other three.