diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index d0dc3f58379a0..9814f92e0739b 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -869,7 +869,8 @@ Other
 - Bug in :meth:`DataFrame.append` that raised ``IndexError`` when appending with empty list (:issue:`28769`)
 - Fix :class:`AbstractHolidayCalendar` to return correct results for
   years after 2030 (now goes up to 2200) (:issue:`27790`)
-- Fixed :class:`IntegerArray` returning ``NA`` rather than ``inf`` for operations dividing by 0 (:issue:`27398`)
+- Fixed :class:`IntegerArray` returning ``inf`` rather than ``NaN`` for operations dividing by 0 (:issue:`27398`)
+- Fixed ``pow`` operations for :class:`IntegerArray` when the other value is ``0`` or ``1`` (:issue:`29997`)
 - Bug in :meth:`Series.count` raises if use_inf_as_na is enabled (:issue:`29478`)
 
 
diff --git a/pandas/core/arrays/integer.py b/pandas/core/arrays/integer.py
index d47e7e3df27e1..3469f782c733a 100644
--- a/pandas/core/arrays/integer.py
+++ b/pandas/core/arrays/integer.py
@@ -718,13 +718,13 @@ def _create_arithmetic_method(cls, op):
         @unpack_zerodim_and_defer(op.__name__)
         def integer_arithmetic_method(self, other):
 
-            mask = None
+            omask = None
 
             if getattr(other, "ndim", 0) > 1:
                 raise NotImplementedError("can only perform ops with 1-d structures")
 
             if isinstance(other, IntegerArray):
-                other, mask = other._data, other._mask
+                other, omask = other._data, other._mask
 
             elif is_list_like(other):
                 other = np.asarray(other)
@@ -742,17 +742,28 @@ def integer_arithmetic_method(self, other):
                     raise TypeError("can only perform ops with numeric values")
 
             # nans propagate
-            if mask is None:
+            if omask is None:
                 mask = self._mask.copy()
             else:
-                mask = self._mask | mask
+                mask = self._mask | omask
 
-            # 1 ** np.nan is 1. So we have to unmask those.
             if op_name == "pow":
-                mask = np.where(self == 1, False, mask)
+                # 1 ** x is 1.
+                mask = np.where((self._data == 1) & ~self._mask, False, mask)
+                # x ** 0 is 1.
+                if omask is not None:
+                    mask = np.where((other == 0) & ~omask, False, mask)
+                else:
+                    mask = np.where(other == 0, False, mask)
 
             elif op_name == "rpow":
-                mask = np.where(other == 1, False, mask)
+                # 1 ** x is 1.
+                if omask is not None:
+                    mask = np.where((other == 1) & ~omask, False, mask)
+                else:
+                    mask = np.where(other == 1, False, mask)
+                # x ** 0 is 1.
+                mask = np.where((self._data == 0) & ~self._mask, False, mask)
 
             with np.errstate(all="ignore"):
                 result = op(self._data, other)
diff --git a/pandas/tests/arrays/test_integer.py b/pandas/tests/arrays/test_integer.py
index d36b42ec87e51..7bb0b065df1da 100644
--- a/pandas/tests/arrays/test_integer.py
+++ b/pandas/tests/arrays/test_integer.py
@@ -346,22 +346,54 @@ def test_divide_by_zero(self, zero, negative):
         result = a / zero
         expected = np.array([np.nan, np.inf, -np.inf, np.nan])
         if negative:
-            values = [np.nan, -np.inf, np.inf, np.nan]
-        else:
-            values = [np.nan, np.inf, -np.inf, np.nan]
-        expected = np.array(values)
+            expected *= -1
         tm.assert_numpy_array_equal(result, expected)
 
-    def test_pow(self):
-        # https://github.com/pandas-dev/pandas/issues/22022
-        a = integer_array([1, np.nan, np.nan, 1])
-        b = integer_array([1, np.nan, 1, np.nan])
+    def test_pow_scalar(self):
+        a = pd.array([0, 1, None, 2], dtype="Int64")
+        result = a ** 0
+        expected = pd.array([1, 1, 1, 1], dtype="Int64")
+        tm.assert_extension_array_equal(result, expected)
+
+        result = a ** 1
+        expected = pd.array([0, 1, None, 2], dtype="Int64")
+        tm.assert_extension_array_equal(result, expected)
+
+        # result = a ** pd.NA
+        # expected = pd.array([None, 1, None, None], dtype="Int64")
+        # tm.assert_extension_array_equal(result, expected)
+
+        result = a ** np.nan
+        expected = np.array([np.nan, 1, np.nan, np.nan], dtype="float64")
+        tm.assert_numpy_array_equal(result, expected)
+
+        # reversed
+        result = 0 ** a
+        expected = pd.array([1, 0, None, 0], dtype="Int64")
+        tm.assert_extension_array_equal(result, expected)
+
+        result = 1 ** a
+        expected = pd.array([1, 1, 1, 1], dtype="Int64")
+        tm.assert_extension_array_equal(result, expected)
+
+        # result = pd.NA ** a
+        # expected = pd.array([1, None, None, None], dtype="Int64")
+        # tm.assert_extension_array_equal(result, expected)
+
+        result = np.nan ** a
+        expected = np.array([1, np.nan, np.nan, np.nan], dtype="float64")
+        tm.assert_numpy_array_equal(result, expected)
+
+    def test_pow_array(self):
+        a = integer_array([0, 0, 0, 1, 1, 1, None, None, None])
+        b = integer_array([0, 1, None, 0, 1, None, 0, 1, None])
         result = a ** b
-        expected = pd.core.arrays.integer_array([1, np.nan, np.nan, 1])
+        expected = integer_array([1, 0, None, 1, 1, 1, 1, None, None])
         tm.assert_extension_array_equal(result, expected)
 
     def test_rpow_one_to_na(self):
         # https://github.com/pandas-dev/pandas/issues/22022
+        # https://github.com/pandas-dev/pandas/issues/29997
         arr = integer_array([np.nan, np.nan])
         result = np.array([1.0, 2.0]) ** arr
         expected = np.array([1.0, np.nan])