Simplify complex numbers worksheet with six algebraic expressions.

To simplify complex number fractions like these, we use multiplication by the complex conjugate of the denominator. This eliminates the imaginary part from the denominator, allowing us to express the result in standard form: \( a + bi \).

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General Method:

Given a fraction \( \frac{a + bi}{c + di} \), multiply numerator and denominator by the conjugate of the denominator: \( c - di \).

Then simplify using:
- Distributive property (FOIL) for numerator
- Difference of squares for denominator: \( (c + di)(c - di) = c^2 + d^2 \)
- Combine real and imaginary parts

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Let’s solve each one step-by-step.

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## 1. \( \frac{8 + 3i}{-3 + i} \)

Conjugate of denominator: \( -3 - i \)

Multiply numerator and denominator:

\[
\frac{(8 + 3i)(-3 - i)}{(-3 + i)(-3 - i)}
\]

Denominator:
\[
(-3)^2 - (i)^2 = 9 - (-1) = 10
\]

Numerator:
\[
(8)(-3) + (8)(-i) + (3i)(-3) + (3i)(-i) = -24 -8i -9i -3i^2
\]
\[
= -24 -17i + 3 \quad (\text{since } i^2 = -1)
\]
\[
= -21 -17i
\]

Result:
\[
\frac{-21 -17i}{10} = \boxed{-\frac{21}{10} - \frac{17}{10}i}
\]

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## 2. \( \frac{9 + 7i}{1 - 3i} \)

Conjugate: \( 1 + 3i \)

\[
\frac{(9 + 7i)(1 + 3i)}{(1 - 3i)(1 + 3i)}
\]

Denominator:
\[
1^2 - (3i)^2 = 1 - (-9) = 10
\]

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

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

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## 3. \( \frac{7 - 7i}{9 - i} \)

Conjugate: \( 9 + i \)

\[
\frac{(7 - 7i)(9 + i)}{(9 - i)(9 + i)}
\]

Denominator:
\[
81 - (-1) = 82
\]

Numerator:
\[
7(9) + 7(i) -7i(9) -7i(i) = 63 + 7i -63i -7i^2
\]
\[
= 63 -56i +7 = 70 -56i
\]

Result:
\[
\frac{70 -56i}{82} = \frac{35 -28i}{41} = \boxed{\frac{35}{41} - \frac{28}{41}i}
\]

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## 4. \( \frac{5 + i}{8 + i} \)

Conjugate: \( 8 - i \)

\[
\frac{(5 + i)(8 - i)}{(8 + i)(8 - i)}
\]

Denominator:
\[
64 - (-1) = 65
\]

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

Result:
\[
\frac{41 +3i}{65} = \boxed{\frac{41}{65} + \frac{3}{65}i}
\]

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## 5. \( \frac{-10 + i}{-1 + 3i} \)

Conjugate: \( -1 - 3i \)

\[
\frac{(-10 + i)(-1 - 3i)}{(-1 + 3i)(-1 - 3i)}
\]

Denominator:
\[
(-1)^2 - (3i)^2 = 1 - (-9) = 10
\]

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

Result:
\[
\frac{13 +29i}{10} = \boxed{\frac{13}{10} + \frac{29}{10}i}
\]

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## 6. \( \frac{-3 -5i}{-8 + 3i} \)

Conjugate: \( -8 - 3i \)

\[
\frac{(-3 -5i)(-8 -3i)}{(-8 + 3i)(-8 -3i)}
\]

Denominator:
\[
64 - (-9) = 73
\]

Numerator:
\[
(-3)(-8) + (-3)(-3i) + (-5i)(-8) + (-5i)(-3i) = 24 +9i +40i +15i^2
\]
\[
= 24 +49i -15 = 9 +49i
\]

Result:
\[
\frac{9 +49i}{73} = \boxed{\frac{9}{73} + \frac{49}{73}i}
\]

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## 7. \( \frac{1 + 2i}{5 + i} \)

Conjugate: \( 5 - i \)

\[
\frac{(1 + 2i)(5 - i)}{(5 + i)(5 - i)}
\]

Denominator:
\[
25 - (-1) = 26
\]

Numerator:
\[
1(5) + 1(-i) + 2i(5) + 2i(-i) = 5 -i +10i -2i^2
\]
\[
= 5 +9i +2 = 7 +9i
\]

Result:
\[
\frac{7 +9i}{26} = \boxed{\frac{7}{26} + \frac{9}{26}i}
\]

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## 8. \( \frac{3 + 3i}{9 - i} \)

Conjugate: \( 9 + i \)

\[
\frac{(3 + 3i)(9 + i)}{(9 - i)(9 + i)}
\]

Denominator:
\[
81 - (-1) = 82
\]

Numerator:
\[
3(9) + 3(i) + 3i(9) + 3i(i) = 27 +3i +27i +3i^2
\]
\[
= 27 +30i -3 = 24 +30i
\]

Result:
\[
\frac{24 +30i}{82} = \frac{12 +15i}{41} = \boxed{\frac{12}{41} + \frac{15}{41}i}
\]

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## ✔ Final Answers:

1. \( \boxed{-\frac{21}{10} - \frac{17}{10}i} \)
2. \( \boxed{-\frac{6}{5} + \frac{17}{5}i} \)
3. \( \boxed{\frac{35}{41} - \frac{28}{41}i} \)
4. \( \boxed{\frac{41}{65} + \frac{3}{65}i} \)
5. \( \boxed{\frac{13}{10} + \frac{29}{10}i} \)
6. \( \boxed{\frac{9}{73} + \frac{49}{73}i} \)
7. \( \boxed{\frac{7}{26} + \frac{9}{26}i} \)
8. \( \boxed{\frac{12}{41} + \frac{15}{41}i} \)

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✔ All simplified to standard form \( a + bi \). Let me know if you’d like to see any steps broken down further!