How would you solve the following exponential equations???

$\displaystyle x^x + y^y = 31$

$\displaystyle x^y + y^x = 17$

answer is2and3. But i want to know the method to solve it.

I will be much thankful to you.

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- Nov 22nd 2008, 10:12 PMShyamsolve exponential equations
How would you solve the following exponential equations???

$\displaystyle x^x + y^y = 31$

$\displaystyle x^y + y^x = 17$

answer is**2**and**3**. But i want to know the method to solve it.

I will be much thankful to you. - Nov 28th 2008, 03:27 AMfardeen_gen
I am trying to solve this problem right from the morning but I am still unsuccessful.(Worried)

Its very simple(as we have the answers) to say:

x^x + y^y = 31 = 4 + 27 = 2^2 + 3^3

x^y + y^x = 17 = 8 + 9 = 2^3 + 3^2

So comparing both sides of the equations, we see that x = 2 and y = 3

Can the MHF helpers please throw some light on what method is to be adopted to solve this type of problem? - Dec 2nd 2008, 05:04 AMfardeen_gen
I have scoured the internet ever since, but no successful discoveries...(Worried)

- Dec 16th 2008, 12:05 PMElliPlease help Shyam
Can somebody please find the answer? (Wait)

- Dec 16th 2008, 12:38 PMIsomorphism
Well we can solve this equation(assuming x and y are positive integers).

Observe that x = 1 has no solution. And by symmetry y = 1 has no solution.

Thus we shall assume $\displaystyle x \geq 2, y \geq 2$.

But $\displaystyle y >3 \Rightarrow y^y > 3^3 = 27 \Rightarrow x^x+ y^y > 31$

Thus y = 2 or 3. Try both of them and get the only possible solutions $\displaystyle (x,y) \in \{(3,2),(2,3)\}$ - Jan 15th 2009, 05:17 PMFreeTSimultaneous Equation?
Considering we have 2 unknowns, and 2 equations, I would imagine that it could be solved with simultaneous equations (eventually).

Here's what I could do:

$\displaystyle x^x+y^y=31$

$\displaystyle x^x=31-y^y$

$\displaystyle x=\sqrt[x]{31-y^y}$

Substituting x into $\displaystyle x^y+y^x=17$ we get:

$\displaystyle \sqrt[x]{31-y^y}^{y}+y^{\sqrt[x]{31-y^y}}=17$

That's as far as I've gotten, I'll think about it more later. If anyone else has any more ideas, please post.

Hope I helped.