Exercises

1. If a and b are real numbers and if ai² = bi⁴, which of the following must be true?

   1. a = b
   2. a = –b
   3. |a| = |b|
   1. I only
   2. II only
   3. III only
   4. II and III only
   5. I, II, and III

2. Which of the following is a negative number?

   1. i²⁵
   2. i⁵⁰
   3. i⁷⁵
   4. i¹⁰⁰
   5. i²⁰⁰

3. Which of the following is equal to (1 + i)²?

   1. –2
   2. 0
   3. 2
   4. –2i
   5. 2i

4. If a and b are real numbers, which of the following is equal to (a + bi)(a – bi)?

   1. a² – b²
   2. a² + b²
   3. a² + 2abi – b²
   4. a² – 2abi – b²
   5. a² – 2abi + b²

5. If a, b, c, and d are consecutive positive integers, what is the value of i^a + i^b + i^c + i^d?

   1. 0
   2. –1
   3. 1
   4. i
   5. It cannot be determined from the information given.

6. If i^m = iⁿ, which of the following must be true?

   1. m = n
   2. m + n is even
   3. n – m is a multiple of 4
   1. I only
   2. III only
   3. I and III only
   4. II and III only
   5. I, II, and III

7. If (a + bi)(c + di) is a real number, which of the following statements must be true?

   1. ac – bd = 0
   2. ac + bd = 0
   3. ad – bc = 0
   4. ad + bc = 0
   5. ab – cd = 0

8. If (2 + 3i) ÷ (3 + 5i) = (a + bi) then what is the value of a + b?

   1. $-{\displaystyle \frac{5}{17}}$
   2. $-{\displaystyle \frac{10}{17}}$
   3. $-{\displaystyle \frac{22}{17}}$
   4. ${\displaystyle \frac{10}{17}}$
   5. ${\displaystyle \frac{22}{17}}$