diff --git a/cirq-core/cirq/ops/clifford_gate.py b/cirq-core/cirq/ops/clifford_gate.py
index 92d6bc65c20..bf7b5ed752c 100644
--- a/cirq-core/cirq/ops/clifford_gate.py
+++ b/cirq-core/cirq/ops/clifford_gate.py
@@ -743,16 +743,21 @@ def _to_phased_xz_gate(self) -> phased_x_z_gate.PhasedXZGate:
         return phased_x_z_gate.PhasedXZGate(x_exponent=x, z_exponent=z, axis_phase_exponent=a)
 
     def __pow__(self, exponent: Union[float, int]) -> 'SingleQubitCliffordGate':
-        # First to check if we can get the sqrt and negative sqrt Clifford.
-        if self._get_sqrt_map().get(exponent, None):
-            pow_gate = self._get_sqrt_map()[exponent].get(self, None)
+        if int(exponent) == exponent:
+            # The single qubit Clifford gates are a group of size 24
+            ret_gate = super().__pow__(int(exponent) % 24)
+            return SingleQubitCliffordGate.from_clifford_tableau(ret_gate.clifford_tableau)
+        elif int(2 * exponent) == 2 * exponent:
+            # If exponent = k/2 for integer k, then we compute the k-th power of the square root
+            if exponent < 0:
+                sqrt_exp = -0.5
+            else:
+                sqrt_exp = 0.5
+            pow_gate = self._get_sqrt_map()[sqrt_exp].get(self, None)
             if pow_gate:
-                return pow_gate
-        # If not, we try the Clifford Tableau based method.
-        ret_gate = super().__pow__(exponent)
-        if ret_gate is NotImplemented:
-            return NotImplemented
-        return SingleQubitCliffordGate.from_clifford_tableau(ret_gate.clifford_tableau)
+                return pow_gate ** (abs(2 * exponent))
+
+        return NotImplemented
 
     def _act_on_(
         self,
diff --git a/cirq-core/cirq/ops/clifford_gate_test.py b/cirq-core/cirq/ops/clifford_gate_test.py
index f3a86b1f321..db0542fee9c 100644
--- a/cirq-core/cirq/ops/clifford_gate_test.py
+++ b/cirq-core/cirq/ops/clifford_gate_test.py
@@ -219,6 +219,8 @@ def test_pow():
     assert cirq.SingleQubitCliffordGate.Y_nsqrt == cirq.SingleQubitCliffordGate.Y**-0.5
     assert cirq.SingleQubitCliffordGate.Z_nsqrt == cirq.SingleQubitCliffordGate.Z**-0.5
     assert cirq.SingleQubitCliffordGate.X_sqrt**-1 == cirq.SingleQubitCliffordGate.X_nsqrt
+    assert cirq.SingleQubitCliffordGate.X_sqrt**3 == cirq.SingleQubitCliffordGate.X**1.5
+    assert cirq.SingleQubitCliffordGate.Z**2.0 == cirq.SingleQubitCliffordGate.I
     assert cirq.inverse(cirq.SingleQubitCliffordGate.X_nsqrt) == (
         cirq.SingleQubitCliffordGate.X_sqrt
     )