​ Express each of the following in a + ib form. 1. (-5i)(i) 2. (-i)(2i)(-i)^3 ​ 3. i^99

Question

Express each of the following in a + ib form. 

1. (-5i)(\(\frac{1}{8}\)i)

2. (-i)(2i)(-\(\frac{1}{8}\)i)³

3. i⁹⁹

4. i¹¹¹ + i²²² + i³³³

5. (7 – i)(2 + 7i)

6. (-1 – i)(4 + 2i)

7. (5 – 3i)²

8. (13 + 3i)³

Answer 1

1. (-5i) (\(\frac{1}{8}\)i) = (-\(\frac{5}{8}\))(i x i) = (-\(\frac{5}{8}\))(-1) = \(\frac{5}{8}\)

 2.

3. i⁹⁹ = i^{96 + 3} = i⁹⁶i³ = -i

4. i¹¹¹ + i²²² + i³³³ + i¹⁰⁸ + i^{220 + 2} + i^{332 + 1}

= i³ + i² + i¹ = -i – 1 + i = -1

5. (7 – i)(2 + 7i) = 7 × 2 – 2i + 7 × 7i – i × 7i

= 14 – 2i + 49i + 7 = 21 + 47i

6. (-1 – i)(4 + 2i) = -4 – 4i – 2i + 2 = – 2 – 6i

7. (5 – 3i)² = 5² – 2 × 5 × 3i + (3i)²

= 25 – 30i – 9 = 16 – 30i

8.