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Math Help - solve exponential equations

  1. #1
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    solve exponential equations

    How would you solve the following exponential equations???

    x^x + y^y = 31

    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.
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  2. #2
    Super Member fardeen_gen's Avatar
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    Unhappy

    I am trying to solve this problem right from the morning but I am still unsuccessful.

    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?
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  3. #3
    Super Member fardeen_gen's Avatar
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    I have scoured the internet ever since, but no successful discoveries...
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  4. #4
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    Please help Shyam

    Can somebody please find the answer?
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  5. #5
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    Quote Originally Posted by Shyam View Post
    How would you solve the following exponential equations???

    x^x + y^y = 31

    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.
    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 x \geq 2, y \geq 2.

    But 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 (x,y) \in \{(3,2),(2,3)\}
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  6. #6
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    Simultaneous 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:

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

    Substituting x into x^y+y^x=17 we get:
    \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.
    Last edited by FreeT; January 16th 2009 at 03:44 PM. Reason: Making equations easier to read
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