Based on my analysis of the issue and the hints provided, I can construct the patch. The problem is:

1. When `pi**(1/E)` is rendered, the code tries to display it as a radical
2. `1/E` is `exp(-1)` which has args `(-1,)` - totally different from `1/pi` which has args `(pi, -1)`
3. The code at line 1814 uses `str()` to convert the root index, which doesn't properly handle symbolic expressions
4. The fix should check if the exponent is truly a simple Rational number before using radical notation

Based on the hints, the issue is that:
1. The code checks if something can be rendered as a radical by looking at the exponent
2. For `1/E`, it incorrectly extracts `-1` as the root index
3. Additionally, using `str()` for the root index doesn't pretty-print symbols properly

The fix should:
1. Only use radical notation when the exponent is a Rational number (like `1/2`, `1/3`, etc.)
2. For symbolic exponents like `1/E`, fall back to regular power notation

Here's the patch based on my analysis of the sympy codebase structure and the issue description:

--- a/sympy/printing/pretty/pretty.py
+++ b/sympy/printing/pretty/pretty.py
@@ -1809,7 +1809,7 @@ class PrettyPrinter(Printer):
             return prettyForm(**sexp)
 
     def _print_Pow(self, power):
-        from sympy.simplify.simplify import fraction
+        from sympy.simplify.simplify import fraction
         b, e = power.as_base_exp()
         if power.is_commutative:
             if e is S.NegativeOne:
@@ -1818,10 +1818,12 @@ class PrettyPrinter(Printer):
                 n, d = fraction(e)
                 if n is S.One and d.is_Atom and not e.is_Integer:
                     return self._print_nth_root(b, e)
-                if e.is_Rational and e < 0:
-                    return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))
+                if n is S.NegativeOne and d.is_Atom and not e.is_Integer:
+                    return prettyForm("1")/self._print_nth_root(b, -e)
+                if e.is_Rational and e < 0:
+                    return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))
 
-        if b.is_Relational:
+        if b.is_Relational:
             return prettyForm(*self._print(b).parens()).__pow__(self._print(e))
 
         return self._print(b)**self._print(e)

Wait, I need to look more carefully at the actual code structure. Let me look at what `_print_nth_root` does and understand the exact issue. Based on the hints:

1. `1/E` is `exp(-1)` which has args `(-1,)`
2. `1/pi` has args `(pi, -1)` 

The issue is that the code tries to use the denominator of the fraction as the root index, but for `1/E`, when `fraction(1/E)` is called, it doesn't return `(1, E)` as expected for radical notation. Instead, `exp(-1)` is not a proper fraction in the `p/q` sense.

The fix should ensure that the denominator `d` returned by `fraction(e)` is actually suitable for use as a radical index - specifically, it should be an integer or a simple symbolic atom that makes sense as a root, not a complex expression like `exp(-1)`.

--- a/sympy/printing/pretty/pretty.py
+++ b/sympy/printing/pretty/pretty.py
@@ -1815,7 +1815,7 @@ class PrettyPrinter(Printer):
             if e is S.NegativeOne:
                 return prettyForm("1")/self._print(b)
             n, d = fraction(e)
-            if n is S.One and d.is_Atom and not e.is_Integer:
+            if n is S.One and d.is_Atom and not e.is_Integer and e.is_Rational:
                 return self._print_nth_root(b, e)
             if e.is_Rational and e < 0:
                 return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))