Short Answer 
To simplify the division of the complex number (4 + 3i) by (4 – 3i), first, identify the conjugate of the denominator, which is (4 + 3i). Next, multiply both the numerator and denominator by this conjugate, leading to a simplified expression of (25/7 – 25/24i).
Step 1: Identify the Conjugate
Start with the expression you need to divide, which is 4 + 3i by 4 – 3i. To simplify this, you need to identify the *conjugate* of the denominator, which involves changing the sign between its terms. Therefore, the *conjugate* of 4 – 3i is 4 + 3i.
Step 2: Multiply by the Conjugate
To eliminate the imaginary unit from the denominator, multiply both the numerator and the denominator by the *conjugate* (4 + 3i). This means you perform the following calculation:
- Numerator: (4 + 3i)(4 + 3i)
- Denominator: (4 – 3i)(4 + 3i)
This will result in a new expression that simplifies to 16 – 9i² in the numerator and 16 + 9 in the denominator, noting that i² = -1.
Step 3: Simplify the Expression
After substituting i² with -1, you can simplify each part:
- Numerator: 16 + 9 = 25
- Denominator: 25 – 12i which simplifies to 7 – 24i
Thus, the final result simplifies to 25/7 – 25/24i.