If the current ratio of the father's age to the son's age is $4:1$ and the product of their ages is $256$, what will be the ratio of their ages after $5$ years?

Let the current age of the father be $4k$ and the current age of the son be $1k$.
Given that the product of their ages is $256$:

$$(4k)\times (k)=256$$

$$4k^2=256$$

$$k^2=64⟹k=8$$

Thus, their current ages are:

* Father's age = $4\times 8=32$ years
* Son's age = $1\times 8=8$ years

Ages after $5$ years will be:

* Father's age = $32+5=37$ years
* Son's age = $8+5=13$ years

The ratio of their ages after $5$ years will be $37:13$.