Can someone help me with factorials and fractional powers?

I'm not sure if this is the right topic or not.

  1. How does fractional powers work? I know #x^3=x*x*x#, but how can you represent #x^(1/2)# without saying #sqrtx# or #x^(a/b)# = #(root(b)(x))^a#, how does #x^(1/2)# give #sqrtx#?

  2. How do fractional and negative factorials work? I know its to do with the Gamma function, where #Gamma(x) = (n-1)!#, or #Gamma(x) = int_0^oo(x^(z-1))(e^-x)dx#, but how does any of these two equations give answers for negative and fractional factorials?

1 Answer
Dec 12, 2017

Answer:

The exponents follow consistent logic. The factorial question has other references. (− n)! is not defined for integers,

Explanation:

The exponents are short-hand for repetitive multiplication or division. As you noted, #x^3 = x xx x xx x#. Similarly, #x^(1/2) = x^(0.5)#.

SO... in analogy with the cubic (or other integer) expression, this is the "product" of halves of the "x". We use the expression "square root" to indicate the inverse of the "square" just as #x^(1/3) = # "Cube root".

The use of the #sqrt# symbol is just another shorthand notation. In practice, for real calculations it is often simpler to use the fractional form instead of the notation. In this way it is easier to do the exponential combinations required. For example,

#sqrt(x) xx .^3sqrt(x) = x^(1/2) xx x^(1/3)#

With the fractional exponent form we can see the solution more easily from the 'rules' of exponential multiplication.

#x^(1/2) xx x^(1/3) = x^(1/2 + 1/3) = x^(5/6)#

That may not be the "nicest" looking form for a value, but it is easily calculated, and much easier to use than the "shorthand" notation form!

A good discussion of the occurrence and use of factorials of negative numbers is here:
http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-30_42.pdf

http://www.multifactorials.com/Graphs%20and%20images/Negative%20factorials.html