# [✓]  Simplify powers, (x^n)^(1/n) to x?

Posted 1 year ago
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 Mathematica fails to simplify the following, even with the Assumptions I give it: In[59]:= FullSimplify[(xe^n)^(1/n), Assumptions -> {xe \[Element] Reals, n \[Element] integers, n > 0}] Out[59]= (xe^n)^(1/n) Does anyone know how to get Mathematica to simplify (x^n)^(1/n) to x?
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Posted 1 year ago
 PowerExpand does it In[2]:= PowerExpand[(x^n)^(1/n)] Out[2]= x 
Posted 1 year ago
 Your simplification of (x^n)^(1/n) to x is correct for x>0, but it is wrong or problematic for x<0, or for complex x. See what happens for x=-2, n=2: With[{x=-2, n=2}, {(x^n)^(1/n), x}] Mathematica refuses to make the simplification unless the user explicitly assumes that x is positive. PowerExpand instructs to ignore all those scruples.
 You are of course right Gianluca. PowerExpand acts here like a quick and easy solution for everyday use. When the variables are real and positive numbers.Given the right assumptions Simplify will also give the expected result: In[1]:= Simplify[(x^n)^(1/n), Assumptions -> {x \[Element] Reals, x > 0, n \[Element] Reals}] Out[1]= x 
 PowerExpand applied to complex numbers: In[1]:= z = a + I b; w = c + I d; In[3]:= u = (z^w)^(1/w) Out[3]= ((a + I b)^(c + I d))^(1/(c + I d)) In[4]:= PowerExpand@u Out[4]= a + I b