If `A diamond B=4A-B`, what is the value of `(3diamond2)diamond3`?

Answer

To solve the problem, we need to evaluate the expression \((3 \diamond 2) \diamond 3\) using the definition of the diamond operation, which is given as \(A \diamond B = 4A - B\).

### Step-by-Step Solution:

1. **Calculate \(3 \diamond 2\)**:
- According to the definition, we substitute \(A = 3\) and \(B = 2\):
\[ 3 \diamond 2 = 4(3) - 2 \]
- Calculate \(4(3)\):
\[ 4(3) = 12 \]
- Now subtract \(2\):
\[ 12 - 2 = 10 \]
- Therefore, \(3 \diamond 2 = 10\).

2. **Now calculate \(10 \diamond 3\)**:
- Again, using the diamond operation definition, substitute \(A = 10\) and \(B = 3\):
\[ 10 \diamond 3 = 4(10) - 3 \]
- Calculate \(4(10)\):
\[ 4(10) = 40 \]
- Now subtract \(3\):
\[ 40 - 3 = 37 \]
- Therefore, \(10 \diamond 3 = 37\).

3. **Final Result**:
- The value of \((3 \diamond 2) \diamond 3\) is:
\[ \boxed{37} \]