# Sympy: Is there a way to apply the collect() function to exponents?

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