Math worksheet for finding the absolute value of complex numbers, featuring ten practice problems.

Let’s solve each problem step by step.

The absolute value of a complex number \( a + bi \) is calculated using the formula:

\[
|a + bi| = \sqrt{a^2 + b^2}
\]

This means we square the real part, square the imaginary part, add them, and take the square root.

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1) |2 - 3i|

Real part: 2 → \(2^2 = 4\)
Imaginary part: -3 → \((-3)^2 = 9\)
Add: \(4 + 9 = 13\)
Square root: \(\sqrt{13}\)

→ Answer: \(\sqrt{13}\)

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2) |-4 + 7i|

Real part: -4 → \((-4)^2 = 16\)
Imaginary part: 7 → \(7^2 = 49\)
Add: \(16 + 49 = 65\)
Square root: \(\sqrt{65}\)

→ Answer: \(\sqrt{65}\)

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3) |-6 + 3i|

Real part: -6 → \((-6)^2 = 36\)
Imaginary part: 3 → \(3^2 = 9\)
Add: \(36 + 9 = 45\)
Square root: \(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\)

→ Answer: \(3\sqrt{5}\)

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4) |-5 + i|

Real part: -5 → \((-5)^2 = 25\)
Imaginary part: 1 → \(1^2 = 1\)
Add: \(25 + 1 = 26\)
Square root: \(\sqrt{26}\)

→ Answer: \(\sqrt{26}\)

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5) |1 - i|

Real part: 1 → \(1^2 = 1\)
Imaginary part: -1 → \((-1)^2 = 1\)
Add: \(1 + 1 = 2\)
Square root: \(\sqrt{2}\)

→ Answer: \(\sqrt{2}\)

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6) |2 + 3i|

Real part: 2 → \(2^2 = 4\)
Imaginary part: 3 → \(3^2 = 9\)
Add: \(4 + 9 = 13\)
Square root: \(\sqrt{13}\)

→ Answer: \(\sqrt{13}\)

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7) |7 - 2i|

Real part: 7 → \(7^2 = 49\)
Imaginary part: -2 → \((-2)^2 = 4\)
Add: \(49 + 4 = 53\)
Square root: \(\sqrt{53}\)

→ Answer: \(\sqrt{53}\)

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8) |1 + i|

Real part: 1 → \(1^2 = 1\)
Imaginary part: 1 → \(1^2 = 1\)
Add: \(1 + 1 = 2\)
Square root: \(\sqrt{2}\)

→ Answer: \(\sqrt{2}\)

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9) |-3 - 4i|

Real part: -3 → \((-3)^2 = 9\)
Imaginary part: -4 → \((-4)^2 = 16\)
Add: \(9 + 16 = 25\)
Square root: \(\sqrt{25} = 5\)

→ Answer: \(5\)

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10) |6 - 7i|

Real part: 6 → \(6^2 = 36\)
Imaginary part: -7 → \((-7)^2 = 49\)
Add: \(36 + 49 = 85\)
Square root: \(\sqrt{85}\)

→ Answer: \(\sqrt{85}\)

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Final Answer:
1) \(\sqrt{13}\)
2) \(\sqrt{65}\)
3) \(3\sqrt{5}\)
4) \(\sqrt{26}\)
5) \(\sqrt{2}\)
6) \(\sqrt{13}\)
7) \(\sqrt{53}\)
8) \(\sqrt{2}\)
9) \(5\)
10) \(\sqrt{85}\)