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April 23rd, 2014, 05:07 AM  #1 
Newbie Joined: Apr 2014 From: new york Posts: 3 Thanks: 0  can anyone solve this??
hello... i wanted to know if anyone can solve this : X^X=64 
April 23rd, 2014, 05:38 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra 
What is the problem you have with it? Without some idea of what you are finding difficult,, it's hard to know how to explain it.

April 23rd, 2014, 05:42 AM  #3 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
You need the Lambert W function $W$ defined as $z=W(z)e^{W(z)}$ for $z\in\mathbb C$. Thus $\displaystyle 64\,=\,x^x$ $\displaystyle \ln64\,=\,x\ln x\,=\,(\ln x)e^{\ln x}$ So $\displaystyle W(\ln64)\,=\,\ln x$ $\displaystyle \implies$ $\displaystyle x\,=\,e^{W(\ln64)}$. 
April 23rd, 2014, 05:44 AM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 
There is no closed form in general. Lambert W function  Wikipedia, the free encyclopedia Scroll down to example 2. I learned this from Mr. TNT (mathbalarka) 
April 23rd, 2014, 05:50 AM  #5 
Newbie Joined: Apr 2014 From: new york Posts: 3 Thanks: 0 
anybody can find that number...it can be a "Square root" or periodic number... the closest one is (√11.556) 