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Introduction to Complex Numbers — Easy Worksheet 1
Grade 11 · MathSheets Name: ____________________ Date: __________
Simplify each expression.
How to solve
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Example
Simplify: i7
Answer: -i
Not quite. The answer is
💡 The powers of i repeat every 4: i1 = i, i2 = -1, i3 = -i and i4 = 1. By definition i2 = -1.
✓ Correct!
Not quite. The answer is
💡 The powers of i repeat every 4: i1 = i, i2 = -1, i3 = -i and i4 = 1. By definition i2 = -1.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i3
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Subtract the real parts and the imaginary parts separately. Real: -4 − (-1) = -3. Imaginary: 4 − 1 = 3. So the answer is -3 + 3i.
✓ Correct!
Not quite. The answer is
💡 Subtract the real parts and the imaginary parts separately. Real: -4 − (-1) = -3. Imaginary: 4 − 1 = 3. So the answer is -3 + 3i.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i6
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply each part of the first number by each part of the second (FOIL): (2 + 2i)(3i) = 6i + 6i2. Since i2 = −1, 6i2 = -6. Real part: 0 − 6 = -6; imaginary part: 6 + 0 = 6. So the product is -6 + 6i.
✓ Correct!
Not quite. The answer is
💡 Multiply each part of the first number by each part of the second (FOIL): (2 + 2i)(3i) = 6i + 6i2. Since i2 = −1, 6i2 = -6. Real part: 0 − 6 = -6; imaginary part: 6 + 0 = 6. So the product is -6 + 6i.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i3
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The powers of i repeat every 4: i1 = i, i2 = -1, i3 = -i and i4 = 1. 8 = 4 × 2, so i8 = (i4)2 = 12 = 1.
✓ Correct!
Not quite. The answer is
💡 The powers of i repeat every 4: i1 = i, i2 = -1, i3 = -i and i4 = 1. 8 = 4 × 2, so i8 = (i4)2 = 12 = 1.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i2
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Add the real parts and the imaginary parts separately. Real: -2 + 3 = 1. Imaginary: 2 + 4 = 6. So the answer is 1 + 6i.
✓ Correct!
Not quite. The answer is
💡 Add the real parts and the imaginary parts separately. Real: -2 + 3 = 1. Imaginary: 2 + 4 = 6. So the answer is 1 + 6i.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i2
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Multiply each part of the first number by each part of the second (FOIL): (-1 + 2i)(3 + i) = -3 − i + 6i + 2i2. Since i2 = −1, 2i2 = -2. Real part: -3 − 2 = -5; imaginary part: -1 + 6 = 5. So the product is -5 + 5i.
✓ Correct!
Not quite. The answer is
💡 Multiply each part of the first number by each part of the second (FOIL): (-1 + 2i)(3 + i) = -3 − i + 6i + 2i2. Since i2 = −1, 2i2 = -2. Real part: -3 − 2 = -5; imaginary part: -1 + 6 = 5. So the product is -5 + 5i.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i8
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The powers of i repeat every 4: i1 = i, i2 = -1, i3 = -i and i4 = 1. 5 = 4 × 1 + 1, so i5 = (i4)1 × i1 = 1 × i1 = i.
✓ Correct!
Not quite. The answer is
💡 The powers of i repeat every 4: i1 = i, i2 = -1, i3 = -i and i4 = 1. 5 = 4 × 1 + 1, so i5 = (i4)1 × i1 = 1 × i1 = i.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i4
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Subtract the real parts and the imaginary parts separately. Real: -2 − (-1) = -1. Imaginary: 1 − 3 = -2. So the answer is -1 - 2i.
✓ Correct!
Not quite. The answer is
💡 Subtract the real parts and the imaginary parts separately. Real: -2 − (-1) = -1. Imaginary: 1 − 3 = -2. So the answer is -1 - 2i.
💡 How to solve: Introduction to Complex Numbers
Powers of i repeat every 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, so divide the exponent by 4 and use the remainder. To add or subtract, combine real parts and imaginary parts separately. To multiply, use FOIL and replace i² with -1, then write the answer as a + bi.
Try a similar one
Simplify: i4
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Introduction to Complex Numbers — Easy Worksheet 1
Simplify each expression.