1. 1. 1. \[ -x^2 + 8x – 3 \]
          
      2. \[ 2a^2 – 18a \]
          
      3. \[ 1 + \frac{3}{a} \]
          
      4. \[ 1 – \frac{4}{b} \]
          
      5. \[ \frac{1}{a} – \frac{1}{b} \]
          
      6. \[ -\frac{2}{b} + \frac{5}{a} \]
          
      7. \[ \frac{4}{5} \]
          
      8. \[ \frac{1}{2} \]
          
      9. \[ 6b^2 \]
          
      10. \[ \frac{6a}{5b} \]
           
      11. \[ -\frac{5x}{24y} \]
           
      12. \[ \frac{5x+6}{10x^2} \]
           
      13. \[ \frac{x(b + 1)}{ab} \]
           
      14. \[ -\frac{7a}{b} \]
           
      15. \[ \frac{1}{3} \]
           
      16. \[ \frac{17}{24} \]
           
      17. \[ \frac{2}{35} \]
           
      18. \[ -\frac{3}{4} \]
           
      19. \[ – \frac{1}{90} \]
           
      20. \[ -\frac{1}{12} \]
           
      21. \[ -7 \]
           
      22. \[ \frac{19}{10} \]
           
      23. \[ 1 \]
           
      24. \[ -\frac{1}{5} \]
           
      25. \[ \frac{3}{2} \]
           
      26. \[ \frac{4}{9} \]
           
      27. \[ \frac{7b}{5a} \]
           
      28. \[ -\frac{5x}{18y} \]
           
      29. \[ 2^{10} = 1.034 \]
           
      30. \[ 1 \]
           
      31. \[ \frac{4}{3} \]
           
      32. \[ \frac{5}{6^2} \]
           
      33. \[ \frac{3}{2^4}=\frac{3}{16} \]
           
      34. \[ \frac{2^6}{3^6} = \frac{64}{729} \]
           
      35. \[ \frac{4}{5} + 1=\frac{9}{5} \]
           
      36. \[ 3 \times 2^3=24 \]
           
      37. \[ \frac{15}{4} \]
           
      38. \[ 3 \times 2^4 = 48 \]
           
      39. \[ 2 \times 5^6 = 31.250 \]
           
      40. \[ \frac{1}{3ab^7} \]
           
      41. \[ \frac{16}{x^2 y^5} \]
           
      42. \[ -\frac{2^7 y^{12}}{x^3} = -128 \frac{y^{12}}{x^3} \]
           
   1. 1. \[ \frac{3^2 a^8}{b^6} = 9\frac{a^8}{b^6} \]
          
   1. \[ \frac{y^{12}}{2^3 x^6} = \frac{y^{12}}{8x^6} \]
       
   2. \[ \frac{1}{4} \]
       
   3. \[ \frac{a^2 b^2}{b^2 – a^2} \]
       
   4. \[ \frac{9x^3 – 2x}{21} \]
       
   5. \[ \frac{3}{4} x^2 y \]
       
   6. \[ \frac{1}{2} \]
       
   7. \[ -\frac{9}{5x} \]
       
   8. 1. \[ 9.44 \times 10^{12}\; km \]
          
      2. \[ 4 \times 10^{-13} \; cm \]
          
      3. \[ 6.251 \times 10^9 \; \text{inhabitants} \]
          
      4. \[ 1.67 \times 10^{-22}\; kg \]
          
   9. 1. \[ 462,400 \; km \]
          
      2. \[ 0.0000000000492 \; m \]
          

Evaluate expressions 1 through 28 and write the answers in their simplest form.

1. \[ 3(x-1)-x(x-5) \]
2. \[ a[5(a-3)-3(a+1)] \]
3. \[ a^{-1}(a+3) \]
4. \[ (2b)^{-1}(2b-8) \]
5. \[ (ab)^{-1}(b-a) \]
6. \[ (-2ab)^{-1}(4a-10b) \]
7. \[ \left( \frac{8}{15} \times \frac{6}{7} \right) \div \frac{4}{7} \]
8. \[ \left( \frac{5a}{12} \times 3b \right) \div \frac{10ab}{4} \]
9. \[ 12ab \div \left( \frac{3a}{2} \times \frac{4}{3b} \right) \]
10. \[ \left( \frac{8a}{5b} \times \frac{a}{10b} \right) \div \frac{2a}{15b} \]
11. \[ \frac{x}{8y} – \frac{x}{3y} \]
12. \[ \frac{3}{5x^2} + \frac{1}{2x} \]
13. \[ \frac{x}{a}+\frac{x}{ab} \]
14. \[ 3\left( \frac{b}{3a} – \frac{2a}{b} \right)- \frac{a^2+b^2}{ab} \]
15. \[ -\frac{5}{21}+1-\frac{3}{7} \]
16. \[ 1-\left( -\frac{3}{8}+\frac{2}{3} \right) \]
17. \[ \left( \frac{1}{5}-2 \right)- \left( \frac{1}{7}-2 \right) \]
18. \[ \frac{9}{10} \left( -\frac{5}{6} \right) \]
19. \[ \left( -\frac{3}{4} \right) \left( -\frac{5}{27} \right) \left( -\frac{2}{25} \right) \]
20. \[ (-5)\left(- \frac{1}{5} \right) \left(\frac{-3}{4}+\frac{-2}{-3} \right) \]
21. \[ \left( \frac{1}{5}-\frac{2}{3} \right) \div \frac{1}{15} \]
22. \[ \left( 4 \div \frac{8}{5} \right)-\left( \frac{9}{10} \div \frac{3}{2} \right) \]
23. \[ \left( \frac{1}{3} + \frac{2}{5} +\frac{1}{30} \right) \div \frac{23}{30} \]
24. \[ \cfrac{\frac{1}{2}-\frac{3}{4}}{\frac{1}{2}+\frac{3}{4}} \]
25. \[ \left( \frac{1}{8}+\frac{3}{4} \right) \left( 1-\frac{5}{12} \right)^{-1} \]
26. \[ \cfrac{\frac{4a}{5}-2a}{\frac{3a}{10}-3a} \]
27. \[ \cfrac{\frac{1}{5a}-\frac{1}{6a}}{\frac{1}{6b}-\frac{1}{7b}} \]
28. \[ \left[ \frac{5}{9a} \div \frac{10}{3a}- \frac{1}{4} \right] \div \left[ \frac{4y}{5x}-\frac{y}{2x} \right] \]

Simplify the expressions from 29 to 50, using positive exponents for all answers.

29. \[ (2^2 \times 2^3)^2 \]
30. \[ \left( \frac{3}{4} \right)^0 \]
31. \[ \left( \frac{3}{4} \right)^{-1} \]
32. \[ 5^{-1}\times \left( \frac{5}{6} \right)^2 \]
33. \[ \left( \frac{2}{3} \right)^{-4} \div 3^{-3} \]
34. \[ \left[ \left( \frac{2}{3} \right)^2 \right]^3 \]
35. \[ \left( \frac{4}{5} \right)^1+\left( \frac{4}{5}\right)^0 \]
36. \[ \frac{3^{-2} \times 3^3}{2^{-3}} \]
37. \[ \left( \frac{2}{3} \right)^{-2}+\left( \frac{2}{3} \right)^{-1} \]
38. \[ \frac{2^3\times3^{-4}\times4^5}{2^5\times3^{-5}\times4^2} \]
39. \[ (5^{-2}\times 2^5)\times(2^{-2} \times 5^4)^{-2} \]
40. \[ \left(3^{}-3a^3b^{-2} \right) \left( 9a^{-4}b^{-5} \right) \]
41. \[ \left( 5x^2y \right)^0 \left( 4^2x^{-2}y^{-5} \right) \]
42. \[ \left( 4x^0 \right)^2 \div \left( -2x^{-1}y^4 \right)^{-3} \]
43. \[ \left( \frac{3a^{-3}b^2}{9ab^{-1}} \right)^{-2} \]
44. \[ \left(\frac{4xy^{-2}}{2x^{-1}y^2} \right)^{-3} \]
45. \[ \left( \frac{5^3m^0n^{-2}}{4^3m^3n^{-5}} \right) \left( \frac{5^3m^{-1}n}{4^2m^2n^{-2}} \right)^{-1} \]
46. \[ \left( a^{-2}-b^{-2} \right)^{-1} \]
47. \[ \frac{3}{7x^{-3}}-\frac{2}{21x^{-1}} \]
48. \[ \left( \frac{x}{2y} \right)^3 \div \left( \frac{4y}{x} \div \frac{2}{3y^3} \right)^{-1} \]
49. \[ \left( \frac{2}{x}+8x^{-1} \right)^{-1} \div 5^{-1}x \]
50. \[ \left(\frac{3}{x^2}-5x^{-2} \right)^{-1} \div \left( \frac{x^3}{9}+\frac{1}{6x^{-3}} \right) \]
51. Write the following numbers using scientific notation.
    1. a light year in kilometers: 9,440,000,000.000 \(km\).
    2. the electron diameter: 0.0000000000004 \(cm\).
    3. the Earth population in 2002: 6,251,000,000 inhabitants.
    4. the neutron mass: 0.000000000000000000000167 \(kg\).
52. Write the following numbers in standard decimal notation:
    1. the distance between the earth to the moon: 4.624 × 105 \(km\).
    2. the length of the X ray wave: 4.92 × 10−11 \(m\).
53. Fraction properties. Prove that:
    1. \[ \frac{ac}{bc}=\frac{a}{b}, c \neq 0 \]
    2. \[ \frac{a}{b}= \frac{c}{d} \Rightarrow ad=bc \]
    3. \[ \frac{-a}{b}= \frac{a}{-b}= -\frac{a}{b} \]
54. Exponent laws. If \(a\) and \(b\) are real numbers; and \(m\), \(n\) are integers, prove that:
    1. \[ \frac{a^n}{a^m}=a^{n-m} \]
    2. \[ \left(a^n \right)^m=a^{nm} \]
    3. \[ \left( ab \right)^n=a^nb^n \]
    4. \[ \frac{a^n}{b^n}=\frac{b^{-n}}{a^{-n}} \]