If the $6$ -digit number $479xyz$ is exactly divisible by $7,11$ and $13$, then the product of the digits $x,y$ and $z$ will be:

Any $6$ -digit number that is exactly divisible by $7,11,$ and $13$ must also be divisible by their product:
$$\text{LCM}(7,11,13)=1001$$
A property of the number $1001$ is that when any $3$ -digit number $abc$ is multiplied by $1001$, it forms a repeating $6$ -digit pattern of the form $abcabc$.
$$abc\times 1001=abcabc$$
Given the number is $479xyz$, it must follow this exact pattern:
$$479xyz=479479$$
By comparing digits:
$$x=4,y=7,z=9$$
The product of the digits $x,y,$ and $z$ is:
$$\text{Product}=4\times 7\times 9=252$$