<s>[INST] Finding the value of $x+y+z$ given sums of two and square root of the other Given: $$x+y+\sqrt{z}=148$$
$$x+\sqrt{y}+z=82$$
$$\sqrt{x}+y+z=98$$
Find the value of $x+y+z$ such that x, y, and z are positive integers.
The only significant step I did was to let $a^2=x, b^2=y, c^2=z.$ After that, I just played with the equations, adding and subtracting, letting $S=x+y+z$, finding equations for S. But I always come to a dead end. Can anyone show me the solution? 
 [/INST] Following on from your step, we subtract the second equation from the third. We then have
$$a-a^2 +b^2-b =16$$ 
or
$$(b-a)(b+a-1)=16$$
No the possible combinations are: $\{1,16\}$, $\{2,8\}$, or $\{4,4\}$. A quick check tells you only the first will work. Thus $b-a=1$ and $b+a-1= 16$. From this we get $a=8$ and $b=9$. Subtracting the third equation from the first we have 
$$
a^2-a + c -c^2=50$$ from which we can solve for $c$ and we obtain $c(c-1)=6$ or $c=3$. Thus $x=64, y =81, z =9.$ 
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