I have the following equation:
$$ f(x,y) = x^{-a} \cdot x^{-b} \cdot y^{a} \cdot y^{-b} \cdot z^{-a} $$
I want to express it as follows:
$$ f(x,y) = \left( \frac{x}{yz}\right)^{a} \cdot \left( \frac{x}{y} \right)^{b} $$
For that I tried the collect() function and the powsimp() function, among others, and I couldn’t get the desired result.
Is it possible to do what I want?
This is my code:
a,b = symbols('a,b', constant = True, positive=True, real=True)
x,y,z = symbols('x,y,z', positive=True, real=True)
eq = x**a * x**b * y**-a * y**-b * z**-a
collect(eq,a, exact = True)
#powsimp(eq)
Returns:
$$ x^{a+b}y^{-a-b}z^{-a} $$
>Solution :
You can do this with the combine='base' argument to powsimp:
In [11]: eq
Out[11]:
a b -a -b -a
x ⋅x ⋅y ⋅y ⋅z
In [12]: powsimp(eq)
Out[12]:
a + b
-a ⎛x⎞
z ⋅⎜─⎟
⎝y⎠
In [13]: powsimp(eq, combine='base')
Out[13]:
b a
⎛x⎞ ⎛ x ⎞
⎜─⎟ ⋅⎜───⎟
⎝y⎠ ⎝y⋅z⎠
In [14]: powsimp(eq, combine='exp')
Out[14]:
a + b -a - b -a
x ⋅y ⋅z
https://docs.sympy.org/latest/modules/simplify/simplify.html#powsimp