Simplify: `(9+4i) ((3)/(2)-i) (9-4i)`

Answer

To simplify the expression \((9 + 4i) \left(\frac{3}{2} - i\right) (9 - 4i)\), we will follow these steps:

### Step 1: Identify the components
We have three parts to multiply: \(9 + 4i\), \(\frac{3}{2} - i\), and \(9 - 4i\).

### Step 2: Use the identity for multiplying conjugates
Notice that \( (9 + 4i)(9 - 4i) \) is a product of conjugates. We can use the identity:
\[ (a + b)(a - b) = a^2 - b^2 \]
Here, \(a = 9\) and \(b = 4i\).

### Step 3: Calculate \( (9 + 4i)(9 - 4i) \)
Using the identity:
\[ (9 + 4i)(9 - 4i) = 9^2 - (4i)^2 \]
Calculating each term:
\[ 9^2 = 81 \quad \text{and} \quad (4i)^2 = 16i^2 = 16(-1) = -16 \]
Thus, we have:
\[ 81 - (-16) = 81 + 16 = 97 \]

### Step 4: Multiply by \(\left(\frac{3}{2} - i\right)\)
Now we need to multiply the result \(97\) by \(\left(\frac{3}{2} - i\right)\):
\[ 97 \left(\frac{3}{2} - i\right) \]
Distributing \(97\):
\[ = 97 \cdot \frac{3}{2} - 97i \]

### Step 5: Calculate \(97 \cdot \frac{3}{2}\)
Calculating the multiplication:
\[ 97 \cdot \frac{3}{2} = \frac{291}{2} \]

### Step 6: Combine the results
Putting it all together, we have:
\[ \frac{291}{2} - 97i \]

### Final Answer
Thus, the simplified form of the expression \((9 + 4i) \left(\frac{3}{2} - i\right) (9 - 4i)\) is:
\[ \frac{291}{2} - 97i \]

---