x@y = `(x + y)^2`
x#y = `(x-y)^2`
Find the value of 12#(14@(12#13)):

Options

A. 4539
B. 42336
C. 45369
D. 89719

Answer

To solve the expression \( 12 \# (14 @ (12 \# 13)) \), we will follow these steps:

### Step 1: Calculate \( 12 \# 13 \)
Using the definition of \( x \# y = (x - y)^2 \):
\[ 12 \# 13 = (12 - 13)^2 = (-1)^2 = 1 \]

### Step 2: Calculate \( 14 @ 1 \)
Using the definition of \( x @ y = (x + y)^2 \):
\[ 14 @ 1 = (14 + 1)^2 = 15^2 = 225 \]

### Step 3: Calculate \( 12 \# 225 \)
Using the definition of \( x \# y = (x - y)^2 \):
\[ 12 \# 225 = (12 - 225)^2 = (-213)^2 = 45369 \]

Thus, the final value of \( 12 \# (14 @ (12 \# 13)) \) is \( 45369 \).

### Final Answer:
\[ \text{The value is } 45369. \]

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