1.7. Solved problems

1. Identify the sets each number is a member of:

     -8    negative composite N-, integer Z, rational Q, real R, complex C
   0.75    rational Q, real R, complex C
     √5    irrational H, real R, complex C
    3/0    not a number (division by zero)
    √-2    complex C
   

2. Simplify: 7 + (2 + 6)²

   = 7 + 8² = 7 + 64 = 71

3. Simplify: {2*[3*(5 + 1) - 2] - 3} - 4

    = {2*[3*6 - 2] - 3} - 4 = 
    = {2*16 - 3} - 4 = 29 - 4 = 25
   

4. Evaluate: (1 + j)(1 - j)

   = 1 - j + j - j² = 2

5. Evaluate: |1 + j|

   = √(1 + 1) = √2

6. Evaluate: i⁰

   = i⁰ = 1

7. Simplify: i¹, i², i³, i⁴, i⁵, i⁶, i⁷, i⁸

   = i, -1, -i, 1, i, -1, -i, 1

8. Simplify: i^-8, i^-7, i^-6, i^-5, i^-4, i^-3, i^-2, i^-1

   = 1, i, -1, -i, 1, i, -1, -i

9. Factor: 360

   = 2×2×2×3×3×5 = 2³×3²×5

10. Factor: 924

    = 2×2×3×7×11 = 2²×3×7×11

11. Evaluate: |2 + 3i|

    = √(4 + 9) = √13

12. Evaluate: (2 + 3i)(2 - 3i)

    = 4 - 6i + 6i - 9i² = 4 + 9 = 13

13. Evaluate: (3 - 4i)²

    = 9 - 12i - 12i + 16i² = 9 - 24i - 16 = -7 - 24i

14. Simplify: 3{[2(2x + 6) + 3x] - 12z}

    = 21x + 36 - 36z