Linked Questions
26 questions linked to/from Fastest way to check if $x^y > y^x$?
107
votes
15
answers
17k
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Comparing $\pi^e$ and $e^\pi$ without calculating them
How can I
compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
- 1,327
19
votes
9
answers
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):
$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
- 1,776
11
votes
9
answers
2k
views
Which of the numbers is larger: $7^{94}$ or $9^{91} $?
In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help.
Which number is larger
$$\begin{...
user321527
17
votes
5
answers
6k
views
Without using a calculator and logarithm, which of $100^{101} , 101^{100}$ is greater?
Which of the following numbers is greater? Without using a calculator and logarithm.
$$100^{101} , 101^{100}$$
My try : $$100=10^2\\101=(100+1)=(10^2+1)$$
So :
$$100^{101}=10^{2(101)}\\101^{100}=...
- 4,782
18
votes
4
answers
7k
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 ...
- 1,269
20
votes
5
answers
5k
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 $20^{30}$ ...
- 1,115
9
votes
9
answers
2k
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 ...
- 1,135
12
votes
6
answers
915
views
Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$
Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$.
My effort: I tried using the fact $9^{\sqrt{2}}<9^{1.5}=27.$
Also We have $512 <729 \Rightarrow 2^9<27^2 \Rightarrow 2^{\frac{9}{...
- 13.2k
15
votes
5
answers
646
views
Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?
Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?
I tried an induction approach. First I showed that if $b=3$ then any $a \geq4$ satisfied $a^b<b^a$.
Then using that ...
- 3,680
7
votes
5
answers
382
views
Which one is bigger: $9^{17}$ and $7^{19}$
One friend asked me to find which one is bigger: $9^{17}$ and $7^{19}$ using basic calculations only. I gave him a solution by using the technique given in here. However, it was not that basic since I ...
- 1,339
4
votes
6
answers
2k
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How would you prove that $2^{50} < 3^{33}$ without directly calculating the values [closed]
Could you generalise the question and get something along the lines of $n^{50} < (n+1)^{33}$ ?
9
votes
5
answers
335
views
$2^{50} < 3^{32}$ using elementary number theory
How would you prove; without big calculations that involve calculator, program or log table; or calculus that
$2^{50} < 3^{32}$
using elementary number theory only?
If it helps you: $2^{50} - ...
- 1,339
6
votes
3
answers
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&...
- 7,975
5
votes
7
answers
315
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?
- 2,044
2
votes
5
answers
183
views
What methods can I use to show that $2^{50} < 3^{33}$, without a calculator
How would I show that $2^{50} < 3^{33}$, without a calculator, and what different methods are there of doing this?
Any help would be much appreciated.
Thanks.
P.S sorry if the tag on this post ...
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