Find the unit digit of the expression $564\times 327\times 819$.

The correct answer is 2.

To find the unit digit of the product of several numbers, we only need to multiply the unit digits (the last digits) of each individual number. The digits in the tens, hundreds, or higher place values do not affect the unit digit of the final product.

First, identify the unit digit of each number in the given expression:
• The unit digit of 564 is 4.
• The unit digit of 327 is 7.
• The unit digit of 819 is 9.

Now, multiply these unit digits step-by-step:

Step 1: Multiply the unit digits of the first two numbers (4 and 7):

$4\times 7=28$

The unit digit of 28 is 8.

Step 2: Multiply this result's unit digit (8) by the unit digit of the third number (9):

$8\times 9=72$

The unit digit of 72 is 2.

Therefore, the unit digit of the given expression

$564\times 327\times 819$

is 2.