If $a=2^x,b=4^y,c=8^z$and $a^{yz}.b^{zx}.c^{xy}=4^6$, then $xyz=$

Given $a=2^x,b=4^y,c=8^z$

$a=2^x$

$b=4^y={\left(2^2\right)}^y=2^{2y}(\because {\left(a^m\right)}^n{a}^{m.n})$

$c=8^z={\left(2^3\right)}^z=2^{3z}$

Substitute a, b, c value in given condition

$a^{yz}.b^{zx}.c^{xy}=4^6$

${\left(2^x\right)}^{yz}.{\left(2^{2y}\right)}^{zx}.{\left(2^{3z}\right)}^{xy}={\left(2^2\right)}^6$

$2^{xyz}{.2}^{2xyz}{.2}^{3xyz}.=2^{12}$

$2^{xyz+2xyz+3xyz}=2^{12}(\because a^m.a^n.a^0=a^{m+n+0})$

$2^{6xyz}=2^{12}\left(\because If{a}^m=a^n\iff m=n\right)$

$6xyz=12$

$xyz=\frac{12}{6}=2$