Complex Number Worksheets (printable, online, answers, examples) - Free Printable

We are given a worksheet titled “Complex Numbers Worksheets” with the instruction: “Simplify.”

Each problem involves multiplying imaginary numbers (multiples of `i`). We must simplify each expression using the key identity:

> i² = -1

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Let’s solve each expression one by one.

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1. `3i • 2i`

Multiply coefficients and `i` terms:

= `(3 × 2) × (i × i)`
= `6 × i²`
= `6 × (-1)`
= -6

✔ Answer: -6

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2. `-5i • -3i`

Multiply coefficients and `i` terms:

= `(-5) × (-3) × (i × i)`
= `15 × i²`
= `15 × (-1)`
= -15

✔ Answer: -15

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3. `-4i • -5i • -2i`

Multiply all coefficients and `i` terms:

Coefficients: `(-4) × (-5) × (-2)` = `20 × (-2)` = -40

`i` terms: `i × i × i` = `i³` = `i² × i` = `(-1) × i` = -i

So total: `-40 × (-i)`? Wait — let's do it step by step:

Actually, better to group:

= `(-4i × -5i) × -2i`
First: `(-4i)(-5i) = 20i² = 20(-1) = -20`
Then: `-20 × -2i = 40i`

Wait — that contradicts above. Let me recalculate carefully.

Actually:

`(-4) × (-5) × (-2) = -40`
`i × i × i = i³ = -i`
So total: `-40 × i³ = -40 × (-i) = 40i`? No — wait:

Hold on: `i³ = i² * i = -1 * i = -i`

So: `-40 × i³ = -40 × (-i) = 40i`

But let’s verify:

Group as: `(-4i * -5i) = 20i² = -20`, then `-20 * -2i = 40i`

✔ Answer: 40i

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4. `2i • -7i • i`

Coefficients: `2 × (-7) × 1 = -14`

`i` terms: `i × i × i = i³ = -i`

Total: `-14 × (-i) = 14i`

Alternatively:

`(2i * -7i) = -14i² = -14*(-1) = 14`, then `14 * i = 14i`

✔ Answer: 14i

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5. `6i • 3i • 5i`

Coefficients: `6 × 3 × 5 = 90`

`i` terms: `i³ = -i`

Total: `90 × (-i) = -90i`

Check:

`(6i * 3i) = 18i² = -18`, then `-18 * 5i = -90i`

✔ Answer: -90i

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6. `i • -i • 3i`

First: `i × -i = -i² = -(-1) = 1`

Then: `1 × 3i = 3i`

✔ Answer: 3i

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7. `-i • -4i`

= `(-1 × i) × (-4 × i)`
= `(-1) × (-4) × i × i`
= `4 × i²`
= `4 × (-1)`
= -4

✔ Answer: -4

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8. `i • i • -7i`

First: `i × i = i² = -1`

Then: `-1 × -7i = 7i`

✔ Answer: 7i

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9. `2i • i`

= `2 × i × i`
= `2 × i²`
= `2 × (-1)`
= -2

✔ Answer: -2

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10. `2i • -2i • 3i • i`

Step-by-step:

Group coefficients: `2 × (-2) × 3 × 1 = -12`

`i` terms: `i × i × i × i = i⁴ = (i²)² = (-1)² = 1`

Total: `-12 × 1 = -12`

Check:

`(2i × -2i) = -4i² = -4(-1) = 4`
`(3i × i) = 3i² = 3(-1) = -3`
Then: `4 × (-3) = -12`

✔ Answer: -12

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## ✔ Final Answers:

| Problem | Simplified Answer |
|---------------------|-------------------|
| 3i • 2i | -6 |
| -5i • -3i | -15 |
| -4i • -5i • -2i | 40i |
| 2i • -7i • i | 14i |
| 6i • 3i • 5i | -90i |
| i • -i • 3i | 3i |
| -i • -4i | -4 |
| i • i • -7i | 7i |
| 2i • i | -2 |
| 2i • -2i • 3i • i | -12 |

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## 📌 Key Concept Recap:
- Always remember: i² = -1
- When multiplying multiple `i`’s: count the number of `i`’s → `i^n`
- `i¹ = i`
- `i² = -1`
- `i³ = -i`
- `i⁴ = 1`
- Then repeats every 4 powers.
- Multiply coefficients separately, then multiply by the power of `i`.

Let me know if you’d like to see visual steps or more examples!