Math worksheet for simplifying radicals using imaginary numbers, featuring problems with square roots and complex number operations.

Let's solve each problem step by step, simplifying radicals using imaginary numbers and performing operations with complex numbers.

---

Key Concepts:

- $\sqrt{-a} = \sqrt{a} \cdot i$, where $i = \sqrt{-1}$
- For complex numbers:
$(a + bi) + (c + di) = (a + c) + (b + d)i$
$(a + bi) - (c + di) = (a - c) + (b - d)i$

---

Problem 1: $\sqrt{-12}$

$$
\sqrt{-12} = \sqrt{12} \cdot i = \sqrt{4 \cdot 3} \cdot i = 2\sqrt{3}i
$$

✔ Answer: $2\sqrt{3}i$

---

Problem 2: $\sqrt{-128}$

$$
\sqrt{-128} = \sqrt{128} \cdot i = \sqrt{64 \cdot 2} \cdot i = 8\sqrt{2}i
$$

✔ Answer: $8\sqrt{2}i$

---

Problem 3: $\sqrt{-512}$

$$
\sqrt{-512} = \sqrt{512} \cdot i = \sqrt{256 \cdot 2} \cdot i = 16\sqrt{2}i
$$

✔ Answer: $16\sqrt{2}i$

---

Problem 4: $\sqrt{-36}$

$$
\sqrt{-36} = \sqrt{36} \cdot i = 6i
$$

✔ Answer: $6i$

---

Problem 5: $\sqrt{-147}$

$$
\sqrt{-147} = \sqrt{147} \cdot i = \sqrt{49 \cdot 3} \cdot i = 7\sqrt{3}i
$$

✔ Answer: $7\sqrt{3}i$

---

Problem 6: $(-5 - 4i) + (8 - 7i)$

Add real parts and imaginary parts:

- Real: $-5 + 8 = 3$
- Imaginary: $-4i - 7i = -11i$

✔ Answer: $3 - 11i$

---

Problem 7: $(-3 - 8i) - (6i + 3i)$

Wait — the second term is written as $(6i + 3i)$, which is just $9i$. But let’s check:
It says: $(-3 - 8i) - (6i + 3i)$ → that’s $(-3 - 8i) - 9i$

Now subtract:

- Real: $-3$
- Imaginary: $-8i - 9i = -17i$

✔ Answer: $-3 - 17i$

> Note: If the problem meant $(-3 - 8i) - (6 + 3i)$, it would be different. But as written, it's $(6i + 3i)$, so we interpret it as $9i$. However, this seems like a typo. Let's double-check.

But looking again:
"(-3 - 8i) - (6i + 3i)" → this is likely intended to be subtraction of two complex numbers. But $(6i + 3i)$ is not standard form. It's probably meant to be $(6 + 3i)$ or $(6i + 3i)$ as $9i$. But if it's truly $(6i + 3i)$, then it's $9i$, so:

$$
(-3 - 8i) - 9i = -3 - 17i
$$

✔ Answer: $-3 - 17i$

---

Problem 8: $(-4 - 5i) - (4 + 6i)$

Subtract:

- Real: $-4 - 4 = -8$
- Imaginary: $-5i - 6i = -11i$

✔ Answer: $-8 - 11i$

---

Problem 9: $(-6 - 7i) + (2 + 6i)$

Add:

- Real: $-6 + 2 = -4$
- Imaginary: $-7i + 6i = -i$

✔ Answer: $-4 - i$

---

Problem 10: $(-5 - 4i) + (8 - 7i)$

This is identical to Problem 6.

- Real: $-5 + 8 = 3$
- Imaginary: $-4i - 7i = -11i$

✔ Answer: $3 - 11i$

---

✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $2\sqrt{3}i$ |
| 2 | $8\sqrt{2}i$ |
| 3 | $16\sqrt{2}i$ |
| 4 | $6i$ |
| 5 | $7\sqrt{3}i$ |
| 6 | $3 - 11i$ |
| 7 | $-3 - 17i$ |
| 8 | $-8 - 11i$ |
| 9 | $-4 - i$ |
| 10 | $3 - 11i$ |

---

Let me know if you'd like these formatted for printing or need explanations in a video-style format!