4.8. Solved problems

1. What is the fourth power of 1.1?

   1.1 × 1.1 × 1.1 × 1.1 = 1.14 = 1.4641
   

2. Rewrite using exponents: x·y·x·x·y·x

   x4y2
   

3. Simplify the term 5x³6x⁴

   30x7
   

4. Simplify the term 2xy²7x³y⁵

   14x4y7
   

5. Simplify the term x^myⁿzⁿx^py^qz^p

   xm+pyn+qzn+p
   

6. Simplify the term a^pb^qc^rab²c³

   ap+1bq+2cr+3
   

7. Expand the term (2x√y)⁴

   16x4y2 
   

8. Expand (x^1/2 + y^1/2)²

   x + 2(xy)1/2 + y 
   

9. Expand (2ab^2/3c^5/3)³

   8a3b2c5
   

10. Expand xy²z³(xyz)⁵

    xy2z3·x5y5z5 = x6y7z8
    

11. Simplify the term (2x/3)^-3

    (3/(2x))3 = 27/(8x3)
    

12. Simplify (x⁵ + 2x⁴ + x³)/x³

    x2 + 2x + 1
    

13. Factor x + 4√x + 4

    (√x + 2)(√x + 2)
    

14. Evaluate 2^-1 + 2^-2 + 2^-3 + 2^-4

    15/16
    

15. Simplify the term -28xy⁵z²/(42x⁵y⁸z)

    2z/(3x4y3)
    

16. Simplify (√x - 1)(√x + 1)

    x - 1
    

17. Simplify (x^1/3 + 1)(x^2/3 - x^1/3 + 1)

    x + 1
    

18. Evaluate (3^-22⁴)^-2

    81/256
    

19. Expand (x^2/3 - 1)³

    x2 - 3x4/3 + 3x2/3 - 1
    

20. Evaluate 121^1/2

    1211/2 = ±√121 = ±11
    

21. Simplify (8x⁶y³)^1/3

    (8x6y3)1/3 = 2x2y
    

22. Simplify (256x⁴y¹⁶)^1/8

    (256x4y16)1/8 = 2y2√x
    

23. Evaluate (1 + i)⁸

    (1 + i)8 = (1 + 2i - 1)4 = (2i)4 = 16i4 = 16
    

24. Use DeMoivre's theorem to evaluate (1 + i)⁸

    (1 + i)8 = {√2[cos(π/4) + sin(π/4)i]}8
             = 24[cos(2π) + sin(2π)i] = 
             = 16[1 + 0i] = 16
    

25. Find the square roots of z = 1 + i

    (1 + i)1/2 = {21/2[cos(π/4) + sin(π/4)i]}1/2, 
                {21/2[cos(π/4 + 2π) + sin(π/4 + 2π)i]}1/2
    
              = 21/4[cos(π/8) + sin(π/8)i], 
                21/4[cos(9π/8) + sin(9π/8)i]
    
              =  [1.0986841 + 0.4550899i], 
                -[1.0986841 + 0.4550899i]
    

26. Evaluate (1 + i)¹⁺ⁱ

    xy = √2·e-π/4·[cos(π/4 + ln √2) + sin(π/4 + ln √2)i]
       = 0.2739573 + 0.5837008i
    

27. Evaluate (2×10³)⁴

    (2×103)4 = 24×1012 = 16×1012 = 1.6×1013
    

28. Evaluate (2×10⁴)^-3

    (2×104)-3 = 2-3×10-12 = 0.125×10-12 = 1.25×10-13
    

29. Proxima Centauri is the star nearest to us; about 4.3 light-years. A light-year is the distance that a photon of light travels in one year at a speed of 300,000 kilometers per second. How far is Proxima Centauri in meters?

    d = 4.3[years] × 365[days/year]
      × 24[hours/day] × 3600[seconds/hour]
      × 300,000[kilometers/second]
      × 1000[meters/kilometer]
      = 4.3 × 3.65×102 × 2.4×10 × 3.6×103 × 3.0×105 × 103
      = (4.3 × 3.65 × 2.4 × 3.6 × 3.0)×102×10×103×105×103
      = 406.8144×1014 = 4.07×1016[meters]
      = 40.7[petameters]
    

30. Evaluate 3²¹ using the squaring algorithm

         result           x     n                     
              1           3    21    start
              3           3    20    result := result * x
              3           9    10    x := x2
              3          81     5    x := x2
            243          81     4    result := result * x
            243        6561     2    x := x2
            243    43046721     1    x := x2
    10460353203    43046721     0    result := result * x
    
     321 = 10460353203
    

    This used seven multiply operations.