diff --git a/pandas/core/computation/expressions.py b/pandas/core/computation/expressions.py
index 8205534c9d48b..957a493925405 100644
--- a/pandas/core/computation/expressions.py
+++ b/pandas/core/computation/expressions.py
@@ -104,11 +104,20 @@ def _evaluate_numexpr(op, op_str, a, b):
         a_value = a
         b_value = b
 
-        result = ne.evaluate(
-            f"a_value {op_str} b_value",
-            local_dict={"a_value": a_value, "b_value": b_value},
-            casting="safe",
-        )
+        try:
+            result = ne.evaluate(
+                f"a_value {op_str} b_value",
+                local_dict={"a_value": a_value, "b_value": b_value},
+                casting="safe",
+            )
+        except TypeError:
+            # numexpr raises eg for array ** array with integers
+            # (https://github.com/pydata/numexpr/issues/379)
+            pass
+
+        if is_reversed:
+            # reverse order to original for fallback
+            a, b = b, a
 
     if _TEST_MODE:
         _store_test_result(result is not None)
diff --git a/pandas/core/ops/array_ops.py b/pandas/core/ops/array_ops.py
index 1d7c16de0c05d..5c26377f44c2b 100644
--- a/pandas/core/ops/array_ops.py
+++ b/pandas/core/ops/array_ops.py
@@ -158,12 +158,14 @@ def _na_arithmetic_op(left, right, op, is_cmp: bool = False):
     try:
         result = expressions.evaluate(op, left, right)
     except TypeError:
-        if is_cmp:
-            # numexpr failed on comparison op, e.g. ndarray[float] > datetime
-            #  In this case we do not fall back to the masked op, as that
-            #  will handle complex numbers incorrectly, see GH#32047
+        if is_object_dtype(left) or is_object_dtype(right) and not is_cmp:
+            # For object dtype, fallback to a masked operation (only operating
+            #  on the non-missing values)
+            # Don't do this for comparisons, as that will handle complex numbers
+            #  incorrectly, see GH#32047
+            result = _masked_arith_op(left, right, op)
+        else:
             raise
-        result = _masked_arith_op(left, right, op)
 
     if is_cmp and (is_scalar(result) or result is NotImplemented):
         # numpy returned a scalar instead of operating element-wise