What is the unit digit in the expansion of $(478749{)}^{(1\times 5\times 2\times 4)}$?

First, find the value of the exponent:
$$\text{Exponent}=1\times 5\times 2\times 4=40$$

The base is $478749$, and its unit digit is $9$. So, we only need to find the unit digit of $9^{40}$.
Let's look at the cyclicity of $9$:

* $9^1=9$
* $9^2=81$ (unit digit is $1$)
* $9^3=729$ (unit digit is $9$)

Thus, $9$ has a cyclicity of $2$:

* If the power is odd, the unit digit is $9$.
* If the power is even, the unit digit is $1$.

Since the exponent $40$ is an even number, the unit digit is $1$.