code: def test_FreeGroupElm__mul__pow__():
      x1 = x.group.dtype(((Symbol("x"), 1),))
      assert x**2 == x1 * x
  
      assert (x**2 * y * x**-2) ** 4 == x**2 * y**4 * x**-2
      assert (x**2) ** 2 == x**4
      assert (x**-1) ** -1 == x
      assert (x**-1) ** 0 == F.identity
      assert (y**2) ** -2 == y**-4
  
      assert x**2 * x**-1 == x
      assert x**2 * y**2 * y**-1 == x**2 * y
      assert x * x**-1 == F.identity
  
      assert x / x == F.identity
      assert x / x**2 == x**-1
      assert (x**2 * y) / (
          x**2 * y**-1
      ) == x**2 * y**2 * x**-2
      assert (x**2 * y) / (
          y**-1 * x**2
      ) == x**2 * y * x**-2 * y
  
      assert x * (x**-1 * y * z * y**-1) == y * z * y**-1
      assert (
          x**2 * (x**-2 * y**-1 * z**2 * y)
          == y**-1 * z**2 * y
      )
  
imports: from sympy.combinatorics.free_groups import free_group, FreeGroup
  from sympy.core import Symbol
  from sympy.testing.pytest import raises
  from sympy.core.numbers import oo
label_desc: Use image node_id 15 for calling a global function with example usage: test_FreeGroupElm__mul__pow__() without return types