If $3^x = 5$, $5^y = 10$, $10^z = 16$, then what is $3^{xyz}$?

Question

Can't post images so I'll type it here:

$$3^x = 5,\qquad 5^y = 10,\qquad 10^z = 16$$

Then what is $3^{xyz}$?

I've spent like an hour trying to solve it and I failed. Help would be super duper appreciated. Thank you!

Edit: uhh I think I solved it? Would the answer be $16$?

Basically I put $3^x$ in place of the $5$ in $5^y = 10$, so now I have $(3^x)^y = 10$ (which is $3^{xy} = 10$), did the same for the last equation and I got $16$ as an answer, but can anyone confirm this?

Answer 1

$3^{xyz}$ is the same as $(3^x)^{yz}$ and $3^x=5$

this becomes $5^{yz}$ and this is the same as $(5^y)^{z}$ and if $5^y=10$

this becomes $10^z$ and since $10^z=16$ you have that $3^{xyz}=16$

Answer 2

Ricky gives the best solution but here is a brute force solution that uses more machinery. We note that $$ x=\frac{\log 5}{\log 3};\quad y=\frac{\log 10}{\log 5}; \quad z=\frac{\log 16}{\log 10} $$ so $$ xyz=\frac{\log 16}{\log 3}=\log_{3}16 $$ whence $$ 3^{xyz}=16. $$