Radical under Radical expression how to find the sum of $\sqrt{\frac54 + \sqrt{\frac32}} + \sqrt{\frac54 - \sqrt{\frac32}} $ ? Is there a method to solve these kind of equations ?

Why does squaring work here? Well it obviously deals with part of the square root. But there are two other things going on.
The first is that $$\left(\sqrt{a+\sqrt b}\right)^2+\left(\sqrt{a-\sqrt b}\right)^2=a+\sqrt b+a-\sqrt b=2a$$
And this means that the inner square roots disappear in part of the square.
The second is that $(a+\sqrt b)(a-\sqrt b)=a^2-b=c^2$. Here the cross term might have left a square root, but $c^2$ has a simple form.
Quite often in questions like this conjugates (expressions with the sign of the square root changed) come into play - the sum and product of two conjugate expressions both tend to be simple. So one technique for such questions is to look for a means to exploit that simplicity.

Note that the idea of conjugates can be generalised considerably, and is one of the ideas behind Galois Theory.
