Question

Can you help me guys?

Answer 1

Since i = √(-1), it follows that i ² = -1, i ³ = -i, and i ⁴ = 1.

q1. We have

i ³¹ = i ²⁸ × i ³ = (i ⁴)⁷ × i ³ = 1⁷ × (-i ) = -i

q2. Approach each square root individually:

√(-20) = √(-1 × 2² × 5) = √(-1) × √(2²) × √5 = 2i √5

√(-12) = √(-1 × 2² × 3) = √(-1) × √(2²) × √3 = 2i √3

Then

√(-20) × √(-12) = (2i √5) × (2i √3) = (2i )² √(5 × 3) = -4√15

You may have been tempted to combine the square roots immediately, but that would have given the wrong answer.

√(-20) × √(-12) ≠ √((-20) × (-12)) = √240 = 4√15

More generally, we have

√a × √b ≠ √(a × b)

for complex numbers a and b. Otherwise, we would have nonsensical claims like 1 = -1 :

√(-1) × √(-1) = i ² = -1

whereas

√(-1) × √(-1) ≠ √((-1)²) = √1 = 1

q3. Nothing tricky about this one:

3i × 4i = 12i ² = -12