December 25th, 2013, 09:18 AM  #1 
Member Joined: Dec 2013 Posts: 86 Thanks: 1  Exponents
Q: The value of (x,y) if and is: (a) (b) (c) (d) ANS: Can this problem be solved algebraically, without plugging in the answer options into the question stem? Request an algebraic solution. 
December 25th, 2013, 11:20 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386 
As x² = y³, y = x^(2/3), and so x^y = (x^(2/3))^x = x^((2/3)x). Hence y = (2/3)x, and so x² = (2/3)³x³. Hence x = (3/2)³ = 27/8, and so y = (2/3)(27/ = 9/4. 
December 26th, 2013, 08:36 PM  #3  
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  Re: Quote:
My solution looks like this x^y = x^(x^(2/3)) = y^x = (x^(2/3))^x = x^(2x/3) so x^(x^(2/3)) = x^(2x/3) so x^(2/3) = 2x/3 so x = 27/8 and y = 9/4 Please clarify. Thanks  
December 27th, 2013, 05:26 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386 
The equation you asked about in my solution is simply a shortened version of the first line in your solution. I substituted for y only in the righthand side of the equation x^y = y^x. 
December 28th, 2013, 09:33 AM  #5 
Member Joined: Dec 2013 Posts: 86 Thanks: 1  Re: Exponents
Hi All ! I have made a solution of my own. Please inform whether it is correct or not. Since so And so Now, since the base is same on both sides = y, we can equate the exponents: or, or, so, And, since Now, please inform whether my solution was correct or wrong! Please inform soon. 
December 28th, 2013, 02:06 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,576 Thanks: 931 Math Focus: Elementary mathematics and beyond  Re: Exponents
That's correct.

December 28th, 2013, 09:55 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386 
There is another solution, but it isn't listed as a possibility. What is it?


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