Helpful math

Properties of exponentials

\(a^{-n} = \frac{1}{a^{n}}\)

>>> a,n = 2,3
>>> print(a**(-n),1.0/a**n)
0.125 0.125

\(a^{1/n} = \sqrt[n]{a}\)

>>> a,n = 2,3
>>> print(a**(1/n), np.cbrt(a))
1.2599210498948732 1.25992104989

\(a^{m/n} = \sqrt[n]{a^{m}}\)

>>> a,m,n = 2,4,3
>>> print(a**(m/n), np.cbrt(a**m))
2.5198420997897464 2.51984209979

\(a^{mn} = (a^{m})^{n}\)

>>> a,m,n = 2,4,3
>>>  print(a**(m*n), (a**m)**n)
4096 4096

\(\log(x^{y}) = y \log(x)\)

>>> x,y = 4,2
>>> print(np.log(x**y),y * np.log(x))
2.77258872224 2.77258872224

\(\log_{a} a^{n} = n\)

>>> x
>>> x
>>> x

\(\log(bc) = \log(b) + \log(c)\)

>>> x
>>> x
>>> x

\(\log(\frac{1}{b}) = -log(b)\)

>>> x
>>> x
>>> x

Also recall that \(n^{0} = 1, \ 1! = 1, \ 0! = 1\)