page_title: Tetration
section_title: Other super-roots
hierarchical_section_title: Tetration / Inverse operations / Super-root / Other super-roots
caption_attribution_description: English: A graph of the cube super root of x, from x=0 to x=6.
context_page_description: In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- and iteration.
  Under the definition as repeated exponentiation, the notation means, where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
  Tetration is also defined recursively as
  ,
  allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.
  The two inverses of tetration are called the super-root and the super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
  Tetration is used for the notation of very large numbers.
context_section_description: For each integer n > 2, the function ⁿx is defined and increasing for x ≥ 1, and ⁿ1 = 1, so that the nth super-root of x, , exists for x ≥ 1.
  However, if the linear approximation above is used, then  if −1 < y ≤ 0, so  cannot exist.
  In the same way as the square super-root, terminology for other super roots can be based on the normal roots: "cube super-roots" can be expressed as ; the "4th super-root" can be expressed as ; and the "nth super-root" is . Note that  may not be uniquely defined, because there may be more than one nᵗʰ root. For example, x has a single (real) super-root if n is odd, and up to two if n is even.
  Just as with the extension of tetration to infinite heights, the super-root can be extended to n = ∞, being well-defined if 1/e ≤ x ≤ e. Note that  and thus that . Therefore, when it is well defined,  and, unlike normal tetration, is an elementary function. For example, .
  It follows from the Gelfond–Schneider theorem that super-root  for any positive integer n is either integer or transcendental, and  is either integer or irrational. It is still an open question whether irrational super-roots are transcendental in the latter case.
caption: The graph 
    
      
        
          y
          =
          
            
              
                x
                
                  3
                
              
            
            
              s
            
          
        
      
      {\displaystyle y={\sqrt[{3}]{x}}_{s}}
    
  .
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