how do you calculate : 3^x+4^x=5^x?

Question

i know you can probably guess the answer, but i would like a mathematical demonstration please!

Answer 1

Don't we need to know what the "^" represents first?

Answer 2

Hi, I'm glad you ask this question. Beautiful!

First I thought I'll just use logarithmic function but I ended up with,

( 3^x + 4^x )^(1/x) = 5 ... eq.1

from here, I'm stuck at solving eq.1 for x.

Now, I let

y = 3^x+4^x-5^x

Then, I plot it, and I found that it intersect at point (2,0), and also, the function is asymptotically stable (converging to y=0) at the negative x-axis.

So this is what I can offer you, with limit (note: inf=infinity)

y = lim(x -> -inf) of (3^x+4^x-5^x)
= 3^(-inf) + 4^(-inf) + 5^(-inf)
= 1/inf + 1/inf + 1/inf = 0 + 0 + 0
= 0

and,

y = lim(x -> 2) of (3^x+4^x-5^x)
= 3^2 + 4^2 - 5^2
= 0

There might be other method to solve this analytically, but numerical method would probably be more effective. Perhaps Newton-Raphson algorithm would do it.

Sorry about that.

Answer 3

This isn't much of a mathematical demonstration, but, if we recognize that 3,4, and 5 are sides of right triangle, we can therefore see that 3^2 + 4^2 = 5^2

x = 2

Answer 4

It is well known that x=2 (Pythogoras theorem)
3, 4 and 5 are pythogoras numbers.

Answer 5

3^x + 4^x = 5^x
-5^x = -5^x

3^x + 4^x - 5^x = 0

7^x - 5^x = 0

2^x = 0

Answer 6

What does ^ mean?
Well, lets see what I can do...
3 times -3 = -9.
-9 + 4 times
First x is -3.
Second x is 3.5
Third x is 0.
Tah - Dah

Answer 7

3^x + 4^x = 5^x

This can only be solved through approximation methods and not algebraically. Hopefully someone can prove me wrong though!

Answer 8

you could always graph the functions.

And go Pythagoras, x=2 is a solution!

3,4,5 family coming back

Answer 9

type it into a calculator:)

Answer 10

x = -21.059