Section 6.1
Positive Integer Exponents
There are five basic Rules of Exponents:
1. ${{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}$
2. ${{(a\cdot b)}^{m}}={{a}^{m}}\cdot {{b}^{m}}$
3. $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
4. ${{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}}$
5. ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
It is important to note that none of these has to do with raising either a sum or a difference to a power.
THERE IS NO GENERAL SIMPLE RULE to help expand either ${{(a+b)}^{m}}$ or${{(a-b)}^{m}}$. Don't invent one!
Examples:
$\text{a. }{{2}^{5}}\cdot {{2}^{7}}={{2}^{5+7}}={{2}^{12}}$ $\text{b. }{{(2x)}^{4}}={{2}^{4}}{{x}^{4}}=16{{x}^{4}}$
$\text{c. }{{\left( {{2}^{3}} \right)}^{4}}={{2}^{3\cdot 4}}={{2}^{12}}$ $\text{d. }{{\left( \frac{2}{3} \right)}^{3}}=\frac{{{2}^{3}}}{{{3}^{3}}}=\frac{8}{27}$
$\text{e. }\frac{{{5}^{7}}}{{{5}^{4}}}={{5}^{7-4}}={{5}^{3}}=125$ $\text{f. }{{\left( 2{{x}^{3}}{{y}^{2}} \right)}^{5}}={{2}^{5}}{{\left( {{x}^{3}} \right)}^{5}}{{\left( {{y}^{2}} \right)}^{5}}={{2}^{5}}{{x}^{15}}{{y}^{10}}$
$\text{g. }\frac{{{8}^{3}}{{x}^{5}}{{y}^{2}}}{{{4}^{4}}{{x}^{2}}{{y}^{2}}}=\frac{{{\left( {{2}^{3}} \right)}^{3}}{{x}^{5}}{{y}^{2}}}{{{\left( {{2}^{2}} \right)}^{4}}{{x}^{2}}{{y}^{2}}}=\frac{{{2}^{9}}{{x}^{5}}{{y}^{2}}}{{{2}^{8}}{{x}^{2}}{{y}^{2}}}=2{{x}^{3}}$ $\text{h. }{{\left( \frac{2x}{3{{y}^{2}}} \right)}^{3}}\cdot {{\left( \frac{9y}{4x} \right)}^{2}}=\frac{8{{x}^{3}}}{27{{y}^{6}}}\cdot \frac{81{{y}^{2}}}{16{{x}^{2}}}=\frac{3x}{2{{y}^{4}}}$
$\text{i. }{{\left( \frac{54{{x}^{2}}}{{{6}^{2}}xy} \right)}^{3}}={{\left( \frac{2\cdot {{3}^{3}}{{x}^{2}}}{{{(2\cdot 3)}^{2}}xy} \right)}^{3}}=\frac{{{2}^{3}}{{\left( {{3}^{3}} \right)}^{3}}{{\left( {{x}^{2}} \right)}^{3}}}{{{2}^{2}}\cdot {{3}^{2}}xy}=\frac{{{2}^{3}}\cdot {{3}^{9}}{{x}^{6}}}{{{2}^{2}}\cdot {{3}^{2}}xy}=\frac{2\cdot {{3}^{7}}{{x}^{5}}}{y}$
$\text{j. }\frac{{{\left( 2{{x}^{2}}y \right)}^{3}}}{4{{x}^{2}}y}=\frac{{{2}^{3}}\cdot {{\left( {{x}^{2}} \right)}^{3}}\cdot {{y}^{3}}}{{{2}^{2}}{{x}^{2}}y}=\frac{{{2}^{3}}{{x}^{6}}{{y}^{3}}}{{{2}^{2}}{{x}^{2}}y}={{2}^{3-2}}\cdot {{x}^{6-2}}\cdot {{y}^{3-1}}=2{{x}^{4}}{{y}^{2}}$
Exponent rules part 2: 2 more exponent rules with an introduction to composite problems
Exercises:
\begin{array}{l l l}
1. \;\;\;{{\left( 3{{x}^{2}} \right)}^{2}}\;\;\; & 2. \;\;\;{{\left( 2{{x}^{3}} \right)}^{3}}\;\;\; & 3. \;\;\;{{2}^{1}}\cdot {{2}^{2}}\cdot {{2}^{3}}\cdot {{2}^{4}}\cdot {{2}^{5}}\;\;\; \\
4. \;\;\;{{x}^{3}}{{y}^{2}}\cdot {{x}^{5}}{{y}^{3}}\;\;\; & 5. \;\;\;x{{y}^{2}}\cdot \left( 2{{x}^{2}}y \right)\;\;\; & 6. \;\;\;x{{y}^{2}}\cdot {{\left( 2{{x}^{2}}y \right)}^{2}}\;\;\; \\
7. \;\;\;{{2}^{5}}{{x}^{6}}{{y}^{7}}\cdot {{2}^{45}}x{{y}^{10}}\;\;\; & 8. \;\;\;{{\left( {{3}^{2}}{{x}^{3}}y \right)}^{3}}\;\;\; & 9. \;\;\;{{\left( {{5}^{12}}{{x}^{4}}{{y}^{2}} \right)}^{12}}\;\;\; \\
10. \;\;\;{{\left( {{10}^{2}}{{a}^{3}}{{b}^{6}} \right)}^{4}}\cdot \left( {{4}^{3}}{{a}^{2}}b \right)\;\;\; & 11. \;\;\;\left( 2x \right){{\left( 2{{x}^{2}} \right)}^{2}}{{\left( 2{{x}^{3}} \right)}^{3}}\;\;\; & 12. \;\;\;\left( 3{{x}^{3}} \right)\cdot {{\left( {{3}^{2}}x{{y}^{2}} \right)}^{2}}\;\;\; \\
13. \;\;\;2{{x}^{3}}\cdot {{2}^{5}}{{x}^{4}}\;\;\; & 14. \;\;\;{{\left( {{2}^{3}}x \right)}^{3}}\cdot {{\left( 2{{x}^{4}} \right)}^{2}}\;\;\; & 15. \;\;\;{{2}^{60}}{{x}^{5}}\cdot {{2}^{15}}{{x}^{2}}\;\;\; \\
16. \;\;\;{{\left( 2{{x}^{3}} \right)}^{2}}\cdot {{\left( {{2}^{5}}{{x}^{4}} \right)}^{3}}\;\;\; & 17. \;\;\;{{3}^{3}}x{{y}^{3}}\cdot {{6}^{5}}{{\left( {{x}^{4}}y \right)}^{2}}\;\;\; & 18. \;\;\;{{\left( 3x{{y}^{3}} \right)}^{3}}\cdot {{\left( 2{{x}^{4}}y \right)}^{3}}\;\;\; \\
19. \;\;\;{{\left( 3{{x}^{3}}y \right)}^{3}}\cdot {{\left( {{3}^{2}}x{{y}^{2}} \right)}^{4}}\;\;\; & 20. \;\;\;{{\left[ 2{{x}^{2}}{{\left( 2x \right)}^{2}} \right]}^{2}}\;\;\; & 21. \;\;\;{{2}^{2x}}\cdot {{2}^{3x}}\cdot {{4}^{4x}}\;\;\; \\
22. \;\;\;\frac{2{{x}^{5}}{{y}^{3}}}{4{{x}^{2}}{{y}^{2}}}\;\;\; & 23. \;\;\;\left( 2{{x}^{2}}{{y}^{3}} \right)\cdot \left( 6{{x}^{3}}y \right)\;\;\; & 24. \;\;\;\frac{{{x}^{3}}{{y}^{2}}}{{{(2xy)}^{3}}}\;\;\; \\
25. \;\;\;\frac{{{\left( 2{{x}^{2}}{{y}^{3}} \right)}^{4}}}{{{x}^{5}}{{y}^{2}}}\;\;\; & 26. \;\;\;\frac{{{2}^{4}}{{x}^{5}}{{y}^{8}}}{{{\left( 2x{{y}^{2}} \right)}^{3}}}\;\;\; & 27. \;\;\;\frac{{{\left( {{2}^{3}}{{x}^{2}}y \right)}^{2}}}{{{4}^{2}}{{x}^{3}}{{y}^{2}}}\;\;\; \\
28. \;\;\;\frac{2\cdot {{3}^{2}}\cdot {{4}^{3}}\cdot {{9}^{4}}}{{{6}^{5}}\cdot {{12}^{3}}}\;\;\; & 29. \;\;\;\frac{{{\left( x{{y}^{2}}{{z}^{3}} \right)}^{4}}}{{{\left( {{x}^{2}}{{y}^{3}}z \right)}^{3}}}\;\;\; & 30. \;\;\;\frac{-{{2}^{10}}{{x}^{13}}{{y}^{4}}}{{{\left( 2{{x}^{2}}y \right)}^{4}}}\;\;\; \\
31. \;\;\;\frac{{{\left( -{{2}^{3}}{{x}^{2}}y \right)}^{3}}}{{{\left( -{{2}^{2}}{{x}^{3}}{{y}^{2}} \right)}^{3}}}\;\;\; & 32. \;\;\;\frac{{{\left( -{{2}^{4}}{{x}^{2}}{{y}^{2}} \right)}^{3}}}{{{\left( -{{2}^{4}}{{x}^{2}}y \right)}^{2}}}\;\;\; & 33. \;\;\;{{x}^{5m}}\cdot {{x}^{3m}}\;\;\; \\
34. \;\;\;{{3}^{m+n}}\cdot {{3}^{m-n}}\;\;\; & 35. \;\;\;{{2}^{m}}\cdot {{3}^{m}}\;\;\; & 36. \;\;\;\frac{{{x}^{5m}}}{{{x}^{3m}}}\;\;\; \\
37. \;\;\;\frac{{{x}^{m+n}}}{{{x}^{m-n}}}\;\;\; & 38. \;\;\;\frac{{{x}^{a+b}}\cdot {{x}^{2a-b}}}{{{x}^{3a}}}\;\;\; & 39. \;\;\;{{\left( 2x \right)}^{m}}\cdot {{\left( 2{{x}^{2}} \right)}^{2m}}\;\;\; \\
40. \;\;\;{{\left( {{x}^{5m}} \right)}^{3}}\;\;\; & 41. \;\;\;{{\left( {{x}^{m+n}} \right)}^{m-n}}\;\;\; & 42. \;\;\;-{{1}^{1}}-{{1}^{2}}-{{1}^{3}}-\ldots -{{1}^{1000}}\;\;\; \\
43. \;\;\;\frac{{{x}^{a+b}}\cdot {{x}^{2a-b}}}{{{\left( {{x}^{a-b}} \right)}^{2}}}\;\;\; & 44. \;\;\;{{\left( {{x}^{m+n}} \right)}^{m+n}}\div \left( {{x}^{{{m}^{2}}+{{n}^{2}}}} \right)\;\;\; & 45. \;\;\;-{{1}^{1}}+{{1}^{2}}-{{1}^{3}}+{{1}^{4}}-\ldots +{{1}^{1000}}\;\;\; \\
46. \;\;\;{{x}^{a}}\cdot {{x}^{b}}\cdot {{y}^{c}}\cdot {{y}^{d}}\;\;\; & 47. \;\;\;{{\left( {{x}^{2a+1}} \right)}^{3a-2}}\;\;\; & 48. \;\;\;{{\left( 2xy \right)}^{a}}\cdot {{\left( 4{{x}^{2}}y \right)}^{2a}}\cdot {{\left( 8x{{y}^{3}} \right)}^{3a}}\;\;\; \\
49. \;\;\;{{x}^{a+b}}\cdot {{x}^{a-b}}\div {{x}^{2}}\;\;\; & 50. \;\;\;{{x}^{{{(a+b)}^{2}}}}\div {{x}^{2ab}}\;\;\; & 51. \;\;\;{{x}^{{{(a+b)}^{2}}}}\div {{x}^{{{(a-b)}^{2}}}}\;\;\;
\end{array}
Identify each of the following as TRUE or FALSE.
\begin{array}{l l}
52. \;\;\;\frac{{{5}^{12}}}{{{5}^{3}}}={{1}^{9}}\;\;\; & 53. \;\;\;{{2}^{2}}\cdot {{3}^{3}}\cdot {{4}^{4}}\cdot {{6}^{6}}={{2}^{16}}\cdot {{3}^{9}}\;\;\; \\
54. \;\;\;\text{If }{{4}^{3}}\cdot {{8}^{5}}={{2}^{x}},\text{ then }x=21.\;\;\; & 55. \;\;\;{{\left( x+3 \right)}^{2}}={{x}^{2}}+9\;\;\; \\
56. \;\;\;{{5}^{m+2}}\cdot {{5}^{m-2}}={{25}^{m}}\;\;\; & 57. \;\;\;{{2}^{m}}+{{2}^{m}}={{2}^{m+1}}\;\;\; \\
58. \;\;\;-{{1}^{1}}-{{1}^{2}}-{{1}^{3}}-\ldots -{{1}^{1000}}=0\;\;\; & 59. \;\;\;{{\left( {{a}^{x}} \right)}^{y}}={{\left( {{a}^{y}} \right)}^{x}}\;\;\; \\
60. \;\;\;{{(5+2)}^{200}}\cdot {{(5-2)}^{200}}={{21}^{200}}\;\;\; & 61. \;\;\;{{2}^{200}}\cdot {{2}^{300}}={{4}^{500}}\;\;\;
\end{array}