<s>[INST] Find $\sqrt a + \sqrt b + \sqrt c$ only in terms of $p$ . 
If :- $$a^2x^3 + b^2y^3 + c^2z^3 = p^5$$
$$ax^2 = by^2 = cz^2$$
$$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{p}$$ Find $\sqrt a  + \sqrt b + \sqrt c$ in terms of $p$ .

What I Tried :- How do you use the information here? I assumed that :-
$$ax^2 = by^2 = cz^2 = k$$
And then I can write the $1$st equation as :- $$k(ax + by + cz) = p^5$$
But I guess that just bring another extra variable into the scene so it does not help .
How can I use the $3$rd equation? I can write it as $\frac{xyz}{xy + yz + zx} = p$ . How do you use it?
 [/INST] Now, for positive variables $a=\frac{k}{x^2},$ $b=\frac{k}{y^2}$, $c=\frac{k}{z^2}$ and  $$a^2x^3+b^2z^3+c^2z^3=k^2\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=p^5,$$ which gives $$k^2=p^6$$ or
$$k=p^3.$$
Id est, $$\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{k}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\sqrt{p}.$$
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