Question: 6+3i write the complex conjugate of the complex number. then multiply the number by its complex conjugate
1. **State the problem:** We are given the complex number $6+3i$.
2. **Find the complex conjugate:** The complex conjugate of a complex number $a+bi$ is $a - bi$.
3. **Apply the formula:** For $6+3i$, the complex conjugate is $6 - 3i$.
4. **Multiply the number by its conjugate:**
$$ (6+3i)(6-3i) $$
5. **Use the distributive property (FOIL):**
$$ 6 \times 6 - 6 \times 3i + 3i \times 6 - 3i \times 3i $$
6. **Simplify each term:**
$$ 36 - 18i + 18i - 9i^2 $$
7. **Combine like terms:** The $-18i$ and $+18i$ cancel out:
$$ 36 - \cancel{18i} + \cancel{18i} - 9i^2 = 36 - 9i^2 $$
8. **Recall that $i^2 = -1$:**
$$ 36 - 9(-1) = 36 + 9 $$
9. **Final result:**
$$ 45 $$
**Answer:** The product of the complex number and its conjugate is $45$.