In a specific mathematical operation represented by the operator @, a consistent rule is followed as shown below:

If $6@4=40$ and $5@3=24$, then determine the missing value in $8@5=?$

The correct answer is 65.

To determine the missing value, we need to analyze the mathematical pattern defined by the operator $@$ using the provided examples.

Step 1: Identify the pattern/rule

Let the operation be expressed as $a@b$, where $a$ is the first number and $b$ is the second number.

By observing the given equations, the pattern followed is:
$a@b=b\times (a+b)$
(or equivalently, $a@b=a\times b+b^2$).

Let's verify this rule with both given cases:

Case 1: For $6@4=40$
$4\times (6+4)=4\times 10=40$
Alternatively: $(6\times 4)+4^2=24+16=40$

Case 2: For $5@3=24$
$3\times (5+3)=3\times 8=24$
Alternatively: $(5\times 3)+3^2=15+9=24$

Step 2: Apply the rule to find the missing value

Now, apply this consistent rule to calculate $8@5$:
$8@5=5\times (8+5)$
$8@5=5\times 13$
$8@5=65$

Thus, the missing value is 65.