Question: Multiply.
$$(12+8i)(12-8i)$$
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Part 1
$$(12+8i)(12-8i)$$ equals
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(Type your answer in the form $a+bi$.)
1. **State the problem:** Multiply the complex numbers $$(12+8i)$$ and $$(12-8i)$$ and express the result in the form $a+bi$.
2. **Recall the formula:** When multiplying two complex numbers in the form $$(a+bi)(c+di)$$, use the distributive property:
$$ (a+bi)(c+di) = ac + adi + bci + bdi^2 $$
Remember that $$i^2 = -1$$.
3. **Apply the formula:**
$$ (12+8i)(12-8i) = 12 \times 12 + 12 \times (-8i) + 8i \times 12 + 8i \times (-8i) $$
4. **Calculate each term:**
$$ 12 \times 12 = 144 $$
$$ 12 \times (-8i) = -96i $$
$$ 8i \times 12 = 96i $$
$$ 8i \times (-8i) = -64i^2 $$
5. **Simplify the expression:**
$$ 144 - 96i + 96i - 64i^2 $$
Notice that $$-96i + 96i = 0$$, so these terms cancel out.
6. **Replace $$i^2$$ with $$-1$$:**
$$ -64i^2 = -64 \times (-1) = 64 $$
7. **Add the real parts:**
$$ 144 + 64 = 208 $$
8. **Final answer:**
$$ 208 + 0i $$
Or simply:
$$ 208 $$