Why does this expression not simplify?

I'm a very beginner with Mathematica, and I do not get why I can't simplify this expression here:

Assuming[
{
a > 0, b > 0, c > 0, d > 0, e > 0, f > 0,
b*e - f^2 > 0,
a*e^2 - b*c*d + (f^2 - b*e)*c > 0
},
FullSimplify[
Sqrt[a*e^2 - b*c*d + (f^2 - b*e)*c]/
Sqrt[-a*e^2 + b*c*d + (-f^2 + b*e)*c]
]
]


In fact, it returns: $$\frac{\sqrt{a e^2-b c (d+e)+c f^2}}{\sqrt{-a e^2+b c (d+e)-c f^2}}$$ which is some kind of reordering of the terms, but not the full simplification. My expected result would be $$-i$$.

I'm assuming that everything is real, and the quantity under square root is positive.

EDIT: I've reproduced a MWE compared to original question, that was much more inricated.

expr = Sqrt[a*e^2 - b*c*d + (f^2 - b*e)*c]/
Sqrt[-a*e^2 + b*c*d + (-f^2 + b*e)*c]


If we allow the indulgence:

expr /. Sqrt[a_]/Sqrt[b_] :> Sqrt[a/b] // FullSimplify


$$i$$

• This gives the wrong result though. The expected answer is $-i$, not $i$
– L.A.
Commented Aug 9, 2023 at 16:25
• Adding an If solves!
– L.A.
Commented Aug 10, 2023 at 9:18
\$Version

(* "13.3.0 for Mac OS X ARM (64-bit) (June 3, 2023)" *)

Clear["Global*"]

Assuming[{
a > 0, b > 0, c > 0, d > 0, e > 0, f > 0,
b*e - f^2 > 0,
a*e^2 - b*c*d + (f^2 - b*e)*c > 0},
Simplify@
ComplexExpand[
Sqrt[a*e^2 - b*c*d + (f^2 - b*e)*c]/
Sqrt[-a*e^2 + b*c*d + (-f^2 + b*e)*c]]]


EDIT: Note, however, that FindInstance cannot find any instances for which the expression is Iwith the given assumptions.

FindInstance[{Sqrt[a*e^2 - b*c*d + (f^2 - b*e)*c]/
Sqrt[-a*e^2 + b*c*d + (-f^2 + b*e)*c] == I, a > 0, b > 0, c > 0, d > 0,
e > 0, f > 0, b*e - f^2 > 0, a*e^2 - b*c*d + (f^2 - b*e)*c > 0}, {a, b, c,
d, e, f}]

(* {} *)


As opposed to,

FindInstance[{Sqrt[a*e^2 - b*c*d + (f^2 - b*e)*c]/
Sqrt[-a*e^2 + b*c*d + (-f^2 + b*e)*c] == -I, a > 0, b > 0, c > 0, d > 0,
e > 0, f > 0, b*e - f^2 > 0, a*e^2 - b*c*d + (f^2 - b*e)*c > 0}, {a, b, c,
d, e, f}, 3]

(* {{a -> 113, b -> 12, c -> 63, d -> 68, e -> 662/29, f -> 379/25}, {a -> 112,
b -> 3, c -> 34, d -> 87, e -> 3/112 (17 + Sqrt[110721]),
f -> 83/20}, {a -> 90, b -> 73, c -> 50, d -> 11, e -> 76, f -> 5}} *)
`