SOLUTION: Values of X satisfying : X ^ sqrt X = ( sqrt X ) ^ X

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Question 1061653: Values of X satisfying : X ^ sqrt X = ( sqrt X ) ^ X
Answer by ikleyn(52915) About Me  (Show Source):
You can put this solution on YOUR website!
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Answer.   The only two values of  x  satisfying this equation  x%5Esqrt%28x%29 = %28sqrt%28x%29%29%5Ex  are  x = 1  and  x = 4.

In other words,  the equation has two and only two solutions  x = 1  and  x = 4.

Proof 

Let consider this equation on its domain x > 0:


x%5Esqrt%28x%29 = %28sqrt%28x%29%29%5Ex.


Take the logarithm of both sides. You will get


sqrt%28x%29%2Aln%28x%29 = x%2Aln%28sqrt%28x%29%29    or


sqrt%28x%29%2Aln%28x%29 = x%2A%281%2F2%29%2Aln%28x%29    or


ln%28x%29%2A%28sqrt%28x%29-+%281%2F2%29%2Ax%29 = 0.


It has two solution:


1.  ln(x) = 0  --->  x = 1,   or/and


2.  sqrt%28x%29 = %281%2F2%29%2Ax  --->  (square both sides)  --->  x = %281%2F4%29%2Ax%5E2  --->  x%5E2 = 4x  --->  x*(x-4) = 0  --->  x = 4

    (x = 0 is not in the domain for the equation (1), and is not under consideration).


So, the statement is proved.

Solved.




Plot y = x%5E%28sqrt%28x%29%29 and y = %28sqrt%28x%29%29%5Ex