Rukevwe wrote: Positive numbers $x$ and $y$ satisfy $xy=2^{15}$ and $\log_2{x} \cdot \log_2{y} = 60$. Find $\sqrt[3]{(\log_2{x})^3+(\log_2{y})^3}$ Fake problem : no such real numbers $x,y$

I think there are, unless I mistyped the problem....

Rukevwe wrote: I think there are, unless I mistyped the problem.... You mistyped. Let $u=\log_2x$ and $v=\log_2y$ Equations are $u+v=15$ and $uv=60$ and so $u,v$ real roots of $X^2-15X+60=0$, whose discriminant is unfortunately negative.

I see what you mean now. But this is actually the correct problem. The solution proposed was: We are looking for the cube root of $u^3+v^3$ Which is $\sqrt[3]{(u+v)^3-3uv(u+v)}$. And this can be determined easily. The solution is correct (I believe), but there are actually no real $x$ and $y$.

Rukevwe wrote: We are looking for the cube root of $u^3+v^3$ Which is $\sqrt[3]{(u+v)^3-3uv(u+v)}$. And this can be determined easily. Yes, but only if $u,v$ exist. Else we can prove any result we want. For example, it is easy to prove $\sqrt[3]{u^3+v^3}=\pi+e^3$