Practice dividing complex numbers and simplifying the results with this free algebra worksheet.

Let's solve each of these complex number division problems by simplifying the expressions. The general method is to multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

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1. $\frac{8 + 3i}{-3 + i}$

Step 1: Multiply numerator and denominator by the conjugate of the denominator: $-3 - i$

$$
\frac{8 + 3i}{-3 + i} \cdot \frac{-3 - i}{-3 - i}
$$

Numerator:
$$
(8 + 3i)(-3 - i) = 8(-3) + 8(-i) + 3i(-3) + 3i(-i) = -24 - 8i - 9i - 3i^2
$$
$$
= -24 - 17i - 3(-1) = -24 - 17i + 3 = -21 - 17i
$$

Denominator:
$$
(-3 + i)(-3 - i) = (-3)^2 - (i)^2 = 9 - (-1) = 9 + 1 = 10
$$

Result:
$$
\frac{-21 - 17i}{10} = -\frac{21}{10} - \frac{17}{10}i
$$

✔ Answer: $-\frac{21}{10} - \frac{17}{10}i$

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2. $\frac{9 + 7i}{1 - 3i}$

Multiply by conjugate $1 + 3i$:

$$
\frac{9 + 7i}{1 - 3i} \cdot \frac{1 + 3i}{1 + 3i}
$$

Numerator:
$$
(9 + 7i)(1 + 3i) = 9(1) + 9(3i) + 7i(1) + 7i(3i) = 9 + 27i + 7i + 21i^2
$$
$$
= 9 + 34i + 21(-1) = 9 + 34i - 21 = -12 + 34i
$$

Denominator:
$$
(1 - 3i)(1 + 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10
$$

Result:
$$
\frac{-12 + 34i}{10} = -\frac{6}{5} + \frac{17}{5}i
$$

✔ Answer: $-\frac{6}{5} + \frac{17}{5}i$

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3. $\frac{7 - 7i}{9 - i}$

Multiply by conjugate $9 + i$:

$$
\frac{7 - 7i}{9 - i} \cdot \frac{9 + i}{9 + i}
$$

Numerator:
$$
(7 - 7i)(9 + i) = 7(9) + 7(i) -7i(9) -7i(i) = 63 + 7i - 63i -7i^2
$$
$$
= 63 - 56i -7(-1) = 63 - 56i + 7 = 70 - 56i
$$

Denominator:
$$
(9 - i)(9 + i) = 81 - i^2 = 81 - (-1) = 82
$$

Result:
$$
\frac{70 - 56i}{82} = \frac{35}{41} - \frac{28}{41}i
$$

✔ Answer: $\frac{35}{41} - \frac{28}{41}i$

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4. $\frac{5 + i}{8 + i}$

Multiply by conjugate $8 - i$:

$$
\frac{5 + i}{8 + i} \cdot \frac{8 - i}{8 - i}
$$

Numerator:
$$
(5 + i)(8 - i) = 5(8) + 5(-i) + i(8) + i(-i) = 40 - 5i + 8i - i^2
$$
$$
= 40 + 3i - (-1) = 40 + 3i + 1 = 41 + 3i
$$

Denominator:
$$
(8 + i)(8 - i) = 64 - i^2 = 64 + 1 = 65
$$

Result:
$$
\frac{41 + 3i}{65} = \frac{41}{65} + \frac{3}{65}i
$$

✔ Answer: $\frac{41}{65} + \frac{3}{65}i$

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5. $\frac{-10 + i}{-1 + 3i}$

Multiply by conjugate $-1 - 3i$:

$$
\frac{-10 + i}{-1 + 3i} \cdot \frac{-1 - 3i}{-1 - 3i}
$$

Numerator:
$$
(-10 + i)(-1 - 3i) = (-10)(-1) + (-10)(-3i) + i(-1) + i(-3i)
= 10 + 30i - i - 3i^2
$$
$$
= 10 + 29i - 3(-1) = 10 + 29i + 3 = 13 + 29i
$$

Denominator:
$$
(-1 + 3i)(-1 - 3i) = (-1)^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10
$$

Result:
$$
\frac{13 + 29i}{10} = \frac{13}{10} + \frac{29}{10}i
$$

✔ Answer: $\frac{13}{10} + \frac{29}{10}i$

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6. $\frac{-3 - 5i}{-8 + 3i}$

Multiply by conjugate $-8 - 3i$:

$$
\frac{-3 - 5i}{-8 + 3i} \cdot \frac{-8 - 3i}{-8 - 3i}
$$

Numerator:
$$
(-3 - 5i)(-8 - 3i) = (-3)(-8) + (-3)(-3i) + (-5i)(-8) + (-5i)(-3i)
= 24 + 9i + 40i + 15i^2
$$
$$
= 24 + 49i + 15(-1) = 24 + 49i - 15 = 9 + 49i
$$

Denominator:
$$
(-8 + 3i)(-8 - 3i) = (-8)^2 - (3i)^2 = 64 - 9(-1) = 64 + 9 = 73
$$

Result:
$$
\frac{9 + 49i}{73} = \frac{9}{73} + \frac{49}{73}i
$$

✔ Answer: $\frac{9}{73} + \frac{49}{73}i$

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7. $\frac{1 + 2i}{5 + i}$

Multiply by conjugate $5 - i$:

$$
\frac{1 + 2i}{5 + i} \cdot \frac{5 - i}{5 - i}
$$

Numerator:
$$
(1 + 2i)(5 - i) = 1(5) + 1(-i) + 2i(5) + 2i(-i) = 5 - i + 10i - 2i^2
$$
$$
= 5 + 9i - 2(-1) = 5 + 9i + 2 = 7 + 9i
$$

Denominator:
$$
(5 + i)(5 - i) = 25 - i^2 = 25 + 1 = 26
$$

Result:
$$
\frac{7 + 9i}{26} = \frac{7}{26} + \frac{9}{26}i
$$

✔ Answer: $\frac{7}{26} + \frac{9}{26}i$

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

Multiply by conjugate $9 + i$:

$$
\frac{3 + 3i}{9 - i} \cdot \frac{9 + i}{9 + i}
$$

Numerator:
$$
(3 + 3i)(9 + i) = 3(9) + 3(i) + 3i(9) + 3i(i) = 27 + 3i + 27i + 3i^2
$$
$$
= 27 + 30i + 3(-1) = 27 + 30i - 3 = 24 + 30i
$$

Denominator:
$$
(9 - i)(9 + i) = 81 - i^2 = 81 + 1 = 82
$$

Result:
$$
\frac{24 + 30i}{82} = \frac{12}{41} + \frac{15}{41}i
$$

✔ Answer: $\frac{12}{41} + \frac{15}{41}i$

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✔ Final Answers:

| Expression | Simplified Form |
|----------|------------------|
| $\frac{8 + 3i}{-3 + i}$ | $-\frac{21}{10} - \frac{17}{10}i$ |
| $\frac{9 + 7i}{1 - 3i}$ | $-\frac{6}{5} + \frac{17}{5}i$ |
| $\frac{7 - 7i}{9 - i}$ | $\frac{35}{41} - \frac{28}{41}i$ |
| $\frac{5 + i}{8 + i}$ | $\frac{41}{65} + \frac{3}{65}i$ |
| $\frac{-10 + i}{-1 + 3i}$ | $\frac{13}{10} + \frac{29}{10}i$ |
| $\frac{-3 - 5i}{-8 + 3i}$ | $\frac{9}{73} + \frac{49}{73}i$ |
| $\frac{1 + 2i}{5 + i}$ | $\frac{7}{26} + \frac{9}{26}i$ |
| $\frac{3 + 3i}{9 - i}$ | $\frac{12}{41} + \frac{15}{41}i$ |

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