Find $\sqrt a + \sqrt b + \sqrt c$ only in terms of $p$ .

Question

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?

Answer 1

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, $$\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{k}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\sqrt{p}.$$

Answer 2

proceeding from your method,$$a^2x^4=b^2y^4=c^2z^4=k^2$$

thus $a^2x^3+b^2y^3+c^2z^3=k^2(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})=p^5$

$k=p^3$

can you end it now?