Find the smallest $n\in \mathbb Z^+$ that makes $\sqrt{100+\sqrt n}+\sqrt{100-\sqrt n}$ integer. Find the smallest $n\in \mathbb Z^+$ that makes $\sqrt{100+\sqrt n}+\sqrt{100-\sqrt n}$
Clearly if $n=0$ then we will have $20$ but I couldn't decide that how can I find the other integers. Any hint?
If I say that $x=\sqrt{100+\sqrt n}+\sqrt{100-\sqrt n}$ then we have $200+2\sqrt{10^4-n}=x^2$

your thinking is absolutely right ,
$x=\sqrt{100+\sqrt n}+\sqrt{100-\sqrt n}$ then we have $200+2\sqrt{10^4-n}=x^2$
for the purposes of this question
$200+2\sqrt{10^4-n}$ has to be perfect square number . and we can also see that x is even
therefore let $g^2=10^4-n$ here $g\in\Bbb Z^+$.
we can also see that $g\leq100$ , therefore $x_{max}=20$
Since $x^2\geq200$
$\implies x\in {\{16,18,20\}}$
$\implies 200+2g \in {\{16^2,18^2,20^2\}}$
$\implies   g \in {\{28,62,100\}}$
$\implies n\in {\{9816,6156,0\}}$
There you go .
