API: Handle pow & rpow special cases

doc/source/user_guide/missing_data.rst

@@ -822,6 +822,18 @@ For example, ``pd.NA`` propagates in arithmetic operations, similarly to
    pd.NA + 1
    "a" * pd.NA
 
+There are a few special cases when the result is known, even when one of the
+operands is ``NA``.
+
+
+================ ======
+Operation        Result
+================ ======
+``pd.NA ** 0``   0
+``1 ** pd.NA``   1
+``-1 ** pd.NA``  -1
+================ ======
+
 In equality and comparison operations, ``pd.NA`` also propagates. This deviates
 from the behaviour of ``np.nan``, where comparisons with ``np.nan`` always
 return ``False``.

pandas/_libs/missing.pyx

@@ -365,8 +365,6 @@ class NAType(C_NAType):
     __rmod__ = _create_binary_propagating_op("__rmod__")
     __divmod__ = _create_binary_propagating_op("__divmod__", divmod=True)
     __rdivmod__ = _create_binary_propagating_op("__rdivmod__", divmod=True)
-    __pow__ = _create_binary_propagating_op("__pow__")
-    __rpow__ = _create_binary_propagating_op("__rpow__")
     # __lshift__ and __rshift__ are not implemented
 
     __eq__ = _create_binary_propagating_op("__eq__")
@@ -383,6 +381,30 @@ class NAType(C_NAType):
     __abs__ = _create_unary_propagating_op("__abs__")
     __invert__ = _create_unary_propagating_op("__invert__")
 
+    # pow has special
+    def __pow__(self, other):
+        if other is C_NA:
+            return NA
+        elif isinstance(other, (numbers.Number, np.bool_)):
+            if other == 0:
+                # returning positive is correct for +/- 0.
+                return type(other)(1)
+            else:
+                return NA
+
+        return NotImplemented
+
+    def __rpow__(self, other):
+        if other is C_NA:
+            return NA
+        elif isinstance(other, (numbers.Number, np.bool_)):
+            if other == 1 or other == -1:
+                return other
+            else:
+                return NA
+
+        return NotImplemented
+
     # Logical ops using Kleene logic
 
     def __and__(self, other):

pandas/tests/scalar/test_na_scalar.py

@@ -38,11 +38,14 @@ def test_arithmetic_ops(all_arithmetic_functions):
     op = all_arithmetic_functions
 
     for other in [NA, 1, 1.0, "a", np.int64(1), np.nan]:
-        if op.__name__ == "rmod" and isinstance(other, str):
+        if op.__name__ in ("pow", "rpow", "rmod") and isinstance(other, str):
             continue
         if op.__name__ in ("divmod", "rdivmod"):
             assert op(NA, other) is (NA, NA)
         else:
+            if op.__name__ == "rpow":
+                # avoid special case
+                other += 1
             assert op(NA, other) is NA
 
 
@@ -69,6 +72,49 @@ def test_comparison_ops():
         assert (other <= NA) is NA
 
 
+@pytest.mark.parametrize(
+    "value",
+    [
+        0,
+        0.0,
+        -0,
+        -0.0,
+        False,
+        np.bool_(False),
+        np.int_(0),
+        np.float_(0),
+        np.int_(-0),
+        np.float_(-0),
+    ],
+)
+def test_pow_special(value):
+    result = pd.NA ** value
+    assert isinstance(result, type(value))
+    assert result == 1
+
+
+@pytest.mark.parametrize(
+    "value",
+    [
+        1,
+        1.0,
+        -1,
+        -1.0,
+        True,
+        np.bool_(True),
+        np.int_(1),
+        np.float_(1),
+        np.int_(-1),
+        np.float_(-1),
+    ],
+)
+def test_rpow_special(value):
+    result = value ** pd.NA
+    assert result == value
+    if not isinstance(value, (np.float_, np.bool_, np.int_)):
+        assert isinstance(result, type(value))
+
+
 def test_unary_ops():
     assert +NA is NA
     assert -NA is NA