Testing that an expression resolves to an integer [closed]

As shown in this video, the expression

$$4 \sqrt{4 - 2 \sqrt{3}} + \sqrt{97 - 56 \sqrt{3}}$$

is in fact an integer.

One way to check is:

FullSimplify[4 Sqrt[4 - 2 Sqrt] + Sqrt[97 - 56 Sqrt]]

which resolves to $$3$$, and

IntegerQ@FullSimplify[4 Sqrt[4 - 2 Sqrt] + Sqrt[97 - 56 Sqrt]]

resolves to True. Great.

But the direct computation:

IntegerQ[4 Sqrt[4 - 2 Sqrt] + Sqrt[97 - 56 Sqrt]]

gives False.

Why doesn't Mathematica yield True to the direct computation? After all, it resolves other such expressions, such as this trivial example:

IntegerQ[Sqrt^2]

(* True *)

• RootReduce can do this simplification and is a pretty powerful tool all-around. Jul 17 '21 at 18:26
• @Roman: Ahh yes... very helpful. Thanks. But still... why would Mathematica give False to the direct approach? Jul 17 '21 at 18:29
• @DavidG.Stork Simplification usually does not happen automatically. Jul 18 '21 at 0:27

The result of

IntegerQ[4 Sqrt[4 - 2 Sqrt] + Sqrt[97 - 56 Sqrt]]

False

is as designed. Let us look in the documentation to IntegerQ, namely, in the Details section

IntegerQ[expr] returns False unless expr is manifestly an integer (i.e. has head Integer).

Let us check

Head[4 Sqrt[4 - 2 Sqrt] + Sqrt[97 - 56 Sqrt]]

Plus