Quiz worksheet titled "Complex Numbers" from Quizizz, featuring 13 multiple-choice questions on simplifying and operating with complex numbers.

Let's solve each problem step by step.

---

Problem 1: Simplify \( (10 + 15i) - (48 - 30i) \)

#### Solution:
1. Distribute the negative sign through the second parenthesis:
\[
(10 + 15i) - (48 - 30i) = 10 + 15i - 48 + 30i
\]

2. Combine the real parts and the imaginary parts separately:
\[
\text{Real part: } 10 - 48 = -38
\]
\[
\text{Imaginary part: } 15i + 30i = 45i
\]

3. Write the simplified result:
\[
-38 + 45i
\]

#### Answer:
\[
\boxed{D}
\]

---

Problem 2: Simplify \( (7 + 2i)(9 - 6i) \)

#### Solution:
1. Use the distributive property (FOIL method):
\[
(7 + 2i)(9 - 6i) = 7 \cdot 9 + 7 \cdot (-6i) + 2i \cdot 9 + 2i \cdot (-6i)
\]

2. Calculate each term:
\[
7 \cdot 9 = 63
\]
\[
7 \cdot (-6i) = -42i
\]
\[
2i \cdot 9 = 18i
\]
\[
2i \cdot (-6i) = -12i^2
\]

3. Combine the terms:
\[
63 - 42i + 18i - 12i^2
\]

4. Simplify the imaginary terms:
\[
-42i + 18i = -24i
\]

5. Recall that \( i^2 = -1 \), so replace \( -12i^2 \) with \( -12(-1) = 12 \):
\[
63 - 24i + 12
\]

6. Combine the real parts:
\[
63 + 12 = 75
\]

7. Write the simplified result:
\[
75 - 24i
\]

#### Answer:
\[
\boxed{C}
\]

---

Problem 3: What does \( i^2 = ? \)

#### Solution:
By definition, the imaginary unit \( i \) is defined as:
\[
i = \sqrt{-1}
\]
Thus:
\[
i^2 = (\sqrt{-1})^2 = -1
\]

#### Answer:
\[
\boxed{D}
\]

---

Problem 4: Simplify \( 3i - (4 + 7i) \)

#### Solution:
1. Distribute the negative sign through the parenthesis:
\[
3i - (4 + 7i) = 3i - 4 - 7i
\]

2. Combine the real parts and the imaginary parts separately:
\[
\text{Real part: } -4
\]
\[
\text{Imaginary part: } 3i - 7i = -4i
\]

3. Write the simplified result:
\[
-4 - 4i
\]

#### Answer:
\[
\boxed{A}
\]

---

Problem 5: Find the sum \( (5 - 2i) + (-7 + 8i) \)

#### Solution:
1. Combine the real parts and the imaginary parts separately:
\[
\text{Real part: } 5 + (-7) = 5 - 7 = -2
\]
\[
\text{Imaginary part: } -2i + 8i = 6i
\]

2. Write the simplified result:
\[
-2 + 6i
\]

#### Answer:
\[
\boxed{B}
\]

---

Problem 6: Multiply \( (2i)(3i) \)

#### Solution:
1. Multiply the coefficients and the imaginary units:
\[
(2i)(3i) = 2 \cdot 3 \cdot i \cdot i = 6i^2
\]

2. Recall that \( i^2 = -1 \):
\[
6i^2 = 6(-1) = -6
\]

#### Answer:
\[
\boxed{A}
\]

---

Final Answers:

1. \(\boxed{D}\)
2. \(\boxed{C}\)
3. \(\boxed{D}\)
4. \(\boxed{A}\)
5. \(\boxed{B}\)
6. \(\boxed{A}\)