Linked Questions

79
votes
11answers
10k views

Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I calculate, without calculator or similar device, the values of $\pi^e$ and $e^\pi$ in order to compare them?
19
votes
9answers
4k views

How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$

How can you determine which one of these numbers is bigger (without calculating): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
17
votes
5answers
4k views

Without using a calculator and logarithm, which of $100^{101} , 101^{100}$ is greater?

Which of the following numbers is greater? Without using a calculator and logarithm. $$100^{101} , 101^{100}$$ My try : $$100=10^2\\101=(100+1)=(10^2+1)$$ So : $$100^{101}=10^{2(101)}\\101^{100}=...
16
votes
4answers
5k views

How to find out which number is larger without a calculator?

So I have a question which is: Which is larger? $$2.2^{3.3} \text{ or } 3.3^{2.2} $$ Now I need to find out with using a calculator but the answer is $3.3^{2.2}$. The only thing I could think of ...
19
votes
5answers
3k views

Is nᵐ>mⁿ if m>n?

I remember playing with my calculator when I was young. I really liked big numbers so I'd punch big numbers like $20^{30}$ to see how big it really is. On such a quest, I did observe that $20^{30}$ ...
7
votes
9answers
787 views

Is $202^{303}$ greater or $303^{202}$?

Find without use of calculator which of the two numbers is greater $202^{303}$ or $303^{202}$. I think we have to do this with calculus because I got this question from my calculus book. I tried ...
6
votes
3answers
1k views

Which is larger, $70^{71}$ or $71^{70}$? [duplicate]

Yet another question of which is larger: $70^{71}$ or $71^{70}$. I solved it by observing that $f(x)=\frac{\ln(x)}{x}$ is decreasing for all $x>e$ since $f'(x)=\frac{1-\ln(x)}{x^2}<0$ for all $x&...
5
votes
7answers
253 views

Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds

I aim to show that $$(n+1)^n<n^{(n+1)}$$ for all $n \geq 3$. I tried induction, but it didn't work. What should I do?
0
votes
3answers
2k views

Unspecified $x^y$ vs. $y^x$ - which is larger?

Given only the expressions $x^y$ and $y^x$ and no additional information except $x\neq y$ (and the meta-knowledge that the problem was presented in the context of induction), is it possible to ...
-1
votes
3answers
184 views

Compare two below natural numbers: $2016^{2017} < 2017^{2016}$ [closed]

Help me Compare the two following natural numbers below $$2016^{2017} < 2017^{2016}?$$ Many thanks.
2
votes
1answer
200 views

Collection of Non-Trick Questions That Require Work to Answer

This question is inspired by a comment on a recent popular question: Which area is larger, the blue area, or the white area? Warning: Spoiler Below! - Don't keep reading if you want to solve this ...
3
votes
1answer
131 views

Which is larger, $e^\pi$ or $\pi^e$? [duplicate]

I don't know how to approach this. I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
1
vote
1answer
54 views

Proving an exponential inequality using calculus

Prove ( using calculus) that $20.17^{20.16}<20.16^{20.17}$. How do I do that?
0
votes
1answer
30 views

A question about exponential functions: $a^b>b^a$ for $b>a>e$ [closed]

How can we prove that for $b>a>e$ ($e$ being the Euler’s number), $a^b$ is greater than $b^a$?