In a certain system of mathematical operations, the operator & symbolises a specific rule such that 2 & 3 = 25 and 3 & 4 = 49. Based on this established pattern, determine the value of 4 & 5.

The correct answer is 81.

To determine the value of 4 & 5, we need to analyze the mathematical pattern established by the operator & in the given examples.

Step 1: Analyze the first given relation (2 & 3 = 25)
First, find the sum of the two numbers:
$2+3=5$
Next, square the resulting sum:
$5^2=25$
This gives the given output of 25.

Step 2: Verify the rule with the second given relation (3 & 4 = 49)
Find the sum of the two numbers:
$3+4=7$
Square the resulting sum:
$7^2=49$
This matches the second output of 49.

From these two steps, we can define the general rule for the operator & between any two numbers $a$ and $b$ as:
$a\&b={(a+b)}^2$

Step 3: Calculate the value for 4 & 5
Apply the rule to $a=4$ and $b=5$:
First, add the two numbers together:
$4+5=9$
Then, square the sum:
$9^2=81$

Therefore, the value of 4 & 5 is 81.