For all numbers a and b, let `a AA b = a^(2)-3ab^(2)`. What is the value of `|5AA(2 AA1)|`?

Answer

To solve the problem step by step, we need to evaluate the expression \( |5 \, AA \, (2 \, AA \, 1)| \) using the defined operation \( a \, AA \, b = a^2 - 3ab^2 \).

### Step 1: Calculate \( 2 \, AA \, 1 \)
Using the operation definition:
\[ 2 \, AA \, 1 = 2^2 - 3 \cdot 2 \cdot 1^2 \]
Calculating each term:
- \( 2^2 = 4 \)
- \( 1^2 = 1 \)
- \( 3 \cdot 2 \cdot 1 = 6 \)

Now substituting these values:
\[ 2 \, AA \, 1 = 4 - 6 = -2 \]

### Step 2: Calculate \( 5 \, AA \, (-2) \)
Now we substitute the result from Step 1 into the next operation:
\[ 5 \, AA \, (-2) = 5^2 - 3 \cdot 5 \cdot (-2)^2 \]
Calculating each term:
- \( 5^2 = 25 \)
- \( (-2)^2 = 4 \)
- \( 3 \cdot 5 \cdot 4 = 60 \)

Now substituting these values:
\[ 5 \, AA \, (-2) = 25 - 60 = -35 \]

### Step 3: Calculate the absolute value
Finally, we need to find the absolute value:
\[ |5 \, AA \, (2 \, AA \, 1)| = |-35| = 35 \]

### Final Answer
Thus, the value of \( |5 \, AA \, (2 \, AA \, 1)| \) is \( \boxed{35} \).

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