How to simplify an expression?

Question

I have two similar expressions. The first expression can be simplified

$3125\ 2^{\frac{1}{2} \left(\sqrt{29}+5\right)} \left(7-\sqrt{29}\right)^{\frac{1}{2} \left(\sqrt{29}-7\right)} \left(\sqrt{29}-5\right)^{\frac{5}{2}-\frac{\sqrt{29}}{2}} \left(\sqrt{29}+3\right)^{-\frac{\sqrt{29}}{2}-\frac{3}{2}}$

FullSimplify[3125 * 2^((1/2) * (5 + Sqrt[29])) * (7 - Sqrt[29])^((1/2)*(-7 + Sqrt[29])) * (-5 + Sqrt[29])^(5/2 - Sqrt[29]/2) * (3 + Sqrt[29])^(-(3/2) - Sqrt[29]/2)]

(* 131 - 22 Sqrt[29] *)

but the second expression not

$27\ 2^{\frac{1}{2} \left(\sqrt{13}+3\right)} \left(5-\sqrt{13}\right)^{\frac{1}{2} \left(\sqrt{13}-5\right)} \left(\sqrt{13}+1\right)^{\frac{1}{2} \left(-\sqrt{13}-1\right)} \left(\sqrt{13}-3\right)^{\frac{3}{2}-\frac{\sqrt{13}}{2}}$

FullSimplify[27 2^(1/2 (3 + Sqrt[13])) (5 - Sqrt[13])^(1/2 (-5 + Sqrt[13])) (1 + Sqrt[13])^(1/2 (-1 - Sqrt[13])) (-3 + Sqrt[13])^(3/2 - Sqrt[13]/2)]

(* 27 2^(1/2 (3 + Sqrt[13])) (5 - Sqrt[13])^(1/2 (-5 + Sqrt[13])) (-3 + Sqrt[13])^(3/2 - Sqrt[13]/2) (1 + Sqrt[13])^(1/2 (-1 - Sqrt[13])) *)

Why ?

The requested result is

(* 1/2 (11 + Sqrt[13]) *)

The numerical check is OK:

N[ 27 2^(1/2 (3 + Sqrt[13])) (5 - Sqrt[13])^(1/2 (-5 + Sqrt[13])) (1 + Sqrt[13])^(1/2 (-1 - Sqrt[13])) (-3 + Sqrt[13])^(3/2 - Sqrt[13]/2), 50]
N[1/2 (11 + Sqrt[13]), 50]

(* 7.3027756377319946465596106337352479731256482869226 *)
(* 7.3027756377319946465596106337352479731256482869226 *)

Answer 1

expr = 27 2^(1/2 (3 + Sqrt[13])) (5 - 
      Sqrt[13])^(1/2 (-5 + Sqrt[13])) (1 + 
      Sqrt[13])^(1/2 (-1 - Sqrt[13])) (-3 + Sqrt[13])^(3/2 - 
      Sqrt[13]/2);

Use ExpToTrig

expr2 = expr // ExpToTrig // FullSimplify

(* (1/2)*(11 + Sqrt[13]) *)

expr == expr2 // FullSimplify

(* True *)

Answer 2

I think you should use RootApproximant because it depends on the properties of the roots

RootApproximant[27 2^(1/2 (3 + Sqrt[13])) (5 - 
          Sqrt[13])^(1/2 (-5 + Sqrt[13])) (1 + 
          Sqrt[13])^(1/2 (-1 - Sqrt[13])) (-3 + Sqrt[13])^(3/2 - 
          Sqrt[13]/2)]

(*  1/2 (11 + Sqrt[13])  *)