conv =>
  lhs
  rw [← Nat.mod_add_div n (ringChar R)]
case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ n = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R)
case neg
R : Type u_1
inst✝¹ : CommSemiring R
n : ℕ
inst✝ : PerfectRing R
x : R
h : ¬ringChar R = 0
⊢ pth_root x ^ (n % ringChar R + ringChar R * (n / ringChar R)) = x ^ (n / ringChar R) * pth_root x ^ (n % ringChar R)