`A DeltaB =A^(B)+B` for all integers A and B. What is the value of `-2Delta(3Delta1)`?

Answer

To solve the problem \( -2 \Delta (3 \Delta 1) \) where the operation \( A \Delta B = A^B + B \), we will follow these steps:

### Step 1: Calculate \( 3 \Delta 1 \)
Using the definition of the operation:
\[ 3 \Delta 1 = 3^1 + 1 \]
Calculating \( 3^1 \):
\[ 3^1 = 3 \]
Now, add 1:
\[ 3 + 1 = 4 \]
So, \( 3 \Delta 1 = 4 \).

### Step 2: Substitute \( 3 \Delta 1 \) into the expression
Now we substitute \( 3 \Delta 1 \) back into the original expression:
\[ -2 \Delta (3 \Delta 1) = -2 \Delta 4 \]

### Step 3: Calculate \( -2 \Delta 4 \)
Using the operation definition again:
\[ -2 \Delta 4 = (-2)^4 + 4 \]
Calculating \( (-2)^4 \):
\[ (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16 \]
Now, add 4:
\[ 16 + 4 = 20 \]

### Final Answer
Thus, the value of \( -2 \Delta (3 \Delta 1) \) is:
\[ \boxed{20} \]