Trac #25218: NumberField attempts to evaluate fractional powers

It still falls back to the symbolic ring, but only if root extraction
within the NumberField fails.

src/sage/rings/number_field/number_field_element.pyx

@@ -2179,8 +2179,12 @@ cdef class NumberFieldElement(FieldElement):
             sage: (1+sqrt2)^-1
             sqrt2 - 1
 
-        If the exponent is not integral, perform this operation in
-        the symbolic ring::
+        If the exponent is not integral, attempt this operation in the NumberField:
+
+            sage: K(2)^(1/2)
+            sqrt2
+
+        If this fails, perform this operation in the symbolic ring::
 
             sage: sqrt2^(1/5)
             2^(1/10)
@@ -2212,30 +2216,39 @@ cdef class NumberFieldElement(FieldElement):
         """
         if (isinstance(base, NumberFieldElement) and
             (isinstance(exp, Integer) or type(exp) is int or exp in ZZ)):
-            return generic_power(base, exp)
-        else:
-            cbase, cexp = canonical_coercion(base, exp)
-            if not isinstance(cbase, NumberFieldElement):
-                return cbase ** cexp
-            # Return a symbolic expression.
-            # We use the hold=True keyword argument to prevent the
-            # symbolics library from trying to simplify this expression
-            # again. This would lead to infinite loops otherwise.
-            from sage.symbolic.ring import SR
+            return generic_power_c(base, exp)
+
+        if (isinstance(base, NumberFieldElement) and exp in QQ):
+            qqexp=QQ(exp)
+            n = qqexp.numerator()
+            d = qqexp.denominator()
             try:
-                res = QQ(base)**QQ(exp)
-            except TypeError:
-                pass
-            else:
-                if res.parent() is not SR:
-                    return parent(cbase)(res)
-                return res
-            sbase = SR(base)
-            if sbase.operator() is operator.pow:
-                nbase, pexp = sbase.operands()
-                return nbase.power(pexp * exp, hold=True)
-            else:
-                return sbase.power(exp, hold=True)
+               return base.nth_root(d)**n
+            except ValueError:
+               pass
+
+        cbase, cexp = canonical_coercion(base, exp)
+        if not isinstance(cbase, NumberFieldElement):
+            return cbase ** cexp
+        # Return a symbolic expression.
+        # We use the hold=True keyword argument to prevent the
+        # symbolics library from trying to simplify this expression
+        # again. This would lead to infinite loops otherwise.
+        from sage.symbolic.ring import SR
+        try:
+            res = QQ(base)**QQ(exp)
+        except TypeError:
+            pass
+        else:
+            if res.parent() is not SR:
+                return parent(cbase)(res)
+            return res
+        sbase = SR(base)
+        if sbase.operator() is operator.pow:
+            nbase, pexp = sbase.operands()
+            return nbase.power(pexp * exp, hold=True)
+        else:
+            return sbase.power(exp, hold=True)
 
     cdef void _reduce_c_(self):
         """