The one with the bigger power will always be bigger. (Works ONLY if the numbers are the same and are just switched places, in other words: if the difference between base and power is equal. Also works only if both of them are greater than e.) 7¹¹ > 11⁷. 63⁶⁴ > 64⁶³. 3^π > π^3. 72818⁹⁹³¹⁸ > 99318⁷²⁸¹⁸. As I said, if atleast one of them isn't greater than e, this wont work. 3² > 2³. But 2.72³ > 3²`⁷², Because 2.72 > e. I don't have a proof of this and I don't know why it works but it just does.
When we compare x^y and y^x if they both positive you can get the 1/x and 1/y power of the both sides and therefore we know compare x^(1/x) and y^(1/y) and if divide both sides with both expressions we start to compare (1/y)^(1/y) and (1/x)^(1/x) and if you look at the function x^x it decreases until 1/e and after that it increases because of that if both x and y are smaller or greater than e you can easily find solution. f(x) = (1/x)^(1/x) for whole numbers f(3) < f(2) = f(4) < f(5) < f(6). And so on. I am not a native English speaker I hope you will understand
meanwhile engineers: both equal 27
So they are equal
3^(22/7) vs (22/7)^3
3^3 • 3^(1/7) vs 22^3 / 7^3
21^3 • 3^(1/7) vs 22^3
3^(1/7) vs (22/21)^3
3 vs (22/21)^21
3 vs (1+1/21)^21
Niece, but pi is approximately 22/7 and pi is not equal to 22/7
The one with the bigger power will always be bigger.
(Works ONLY if the numbers are the same and are just switched places, in other words: if the difference between base and power is equal. Also works only if both of them are greater than e.)
7¹¹ > 11⁷.
63⁶⁴ > 64⁶³.
3^π > π^3.
72818⁹⁹³¹⁸ > 99318⁷²⁸¹⁸.
As I said, if atleast one of them isn't greater than e, this wont work.
3² > 2³.
But 2.72³ > 3²`⁷²,
Because 2.72 > e.
I don't have a proof of this and I don't know why it works but it just does.
Check the link for the proof
ruclips.net/video/PImaOMEfsXw/видео.html
When we compare x^y and y^x if they both positive you can get the 1/x and 1/y power of the both sides and therefore we know compare x^(1/x) and y^(1/y) and if divide both sides with both expressions we start to compare (1/y)^(1/y) and (1/x)^(1/x) and if you look at the function x^x it decreases until 1/e and after that it increases because of that if both x and y are smaller or greater than e you can easily find solution. f(x) = (1/x)^(1/x) for whole numbers f(3) < f(2) = f(4) < f(5) < f(6). And so on. I am not a native English speaker I hope you will understand