Complex numbers
Answers are for parents and teachers.
1
Simplify √(−64) and √(−50).
√(−64) = 8i. √(−50) = √50 × i = 5√2 i.
2
Find i⁸³ + i⁻³.
83 = 80 + 3, so i⁸³ = i³ = −i. −3 = −4 + 1, so i⁻³ = i. The sum is −i + i = 0.
3
Solve 5x² + 80 = 0 and 2x² + 9 = 1.
5x² = −80, x² = −16, x = ±4i. 2x² = −8, x² = −4, x = ±2i.
4
Find x and y if (3x − 2y) + (x + y)i = 4 + 3i.
3x − 2y = 4 and x + y = 3. So y = 3 − x and 3x − 6 + 2x = 4, so x = 2 and y = 1. Check: 6 − 2 = 4 ✓, 2 + 1 = 3 ✓.
5
Calculate (4 + i)(3 − 2i) and (2 + 6i)/(1 + i).
(4 + i)(3 − 2i) = 12 − 8i + 3i − 2i² = 14 − 5i. For the quotient multiply by 1 − i: top (2 + 6i)(1 − i) = 2 − 2i + 6i − 6i² = 8 + 4i, bottom 2, so the answer is 4 + 2i.