Fillable Online Conjugates and Dividing Complex Numbers ... - Free Printable

Let's solve the worksheet step by step.

---

Understanding the Concepts

#### 1. Complex Conjugate
The conjugate of a complex number $ a + bi $ is $ a - bi $.
- You change the sign of the imaginary part.

#### 2. Dividing Complex Numbers
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

---

## Problems 1–6: Find the Conjugates

We simply change the sign of the imaginary part.

1. $-10i$
→ This is $ 0 - 10i $, so conjugate is $ 0 + 10i = \boxed{10i} $

2. $-3i$
→ $ 0 - 3i $ → conjugate is $ \boxed{3i} $

3. $-6i$
→ $ 0 - 6i $ → conjugate is $ \boxed{6i} $

4. $2 - 6i$
→ conjugate is $ \boxed{2 + 6i} $

5. $8 - 5i$
→ conjugate is $ \boxed{8 + 5i} $

6. $3 - 9i$
→ conjugate is $ \boxed{3 + 9i} $

---

## Problems 7–10: Simplify and Complete Operations

We simplify fractions involving complex numbers by multiplying numerator and denominator by the conjugate of the denominator.

---

7. $6 - 12i$

This is just a complex number — no operation needed. But the instruction says "simplify and complete any necessary operations". Since it’s already simplified:

→ $ \boxed{6 - 12i} $

---

8. $\frac{4}{7 + 2i}$

Multiply numerator and denominator by the conjugate of the denominator: $7 - 2i$

$$
\frac{4}{7 + 2i} \cdot \frac{7 - 2i}{7 - 2i} = \frac{4(7 - 2i)}{(7 + 2i)(7 - 2i)}
$$

Numerator:
$4(7 - 2i) = 28 - 8i$

Denominator:
Use difference of squares:
$(7)^2 - (2i)^2 = 49 - 4i^2 = 49 - 4(-1) = 49 + 4 = 53$

So:
$$
\frac{28 - 8i}{53} = \boxed{\frac{28}{53} - \frac{8}{53}i}
$$

---

9. $\frac{8}{3 + 6i}$

Multiply numerator and denominator by the conjugate: $3 - 6i$

$$
\frac{8}{3 + 6i} \cdot \frac{3 - 6i}{3 - 6i} = \frac{8(3 - 6i)}{(3 + 6i)(3 - 6i)}
$$

Numerator:
$8(3 - 6i) = 24 - 48i$

Denominator:
$3^2 - (6i)^2 = 9 - 36i^2 = 9 - 36(-1) = 9 + 36 = 45$

So:
$$
\frac{24 - 48i}{45} = \frac{24}{45} - \frac{48}{45}i = \boxed{\frac{8}{15} - \frac{16}{15}i}
$$

(Simplified: divide numerator and denominator by 3)

---

10. $\frac{5}{9 + 4i}$

Multiply numerator and denominator by the conjugate: $9 - 4i$

$$
\frac{5}{9 + 4i} \cdot \frac{9 - 4i}{9 - 4i} = \frac{5(9 - 4i)}{(9 + 4i)(9 - 4i)}
$$

Numerator:
$5(9 - 4i) = 45 - 20i$

Denominator:
$9^2 - (4i)^2 = 81 - 16i^2 = 81 - 16(-1) = 81 + 16 = 97$

So:
$$
\frac{45 - 20i}{97} = \boxed{\frac{45}{97} - \frac{20}{97}i}
$$

---

## ✔ Final Answers:

Conjugates (1–6):

1. $ \boxed{10i} $
2. $ \boxed{3i} $
3. $ \boxed{6i} $
4. $ \boxed{2 + 6i} $
5. $ \boxed{8 + 5i} $
6. $ \boxed{3 + 9i} $

Simplifications (7–10):

7. $ \boxed{6 - 12i} $
8. $ \boxed{\frac{28}{53} - \frac{8}{53}i} $
9. $ \boxed{\frac{8}{15} - \frac{16}{15}i} $
10. $ \boxed{\frac{45}{97} - \frac{20}{97}i} $

---

Let me know if you'd like these answers formatted for printing or need explanations for any specific step!