数列与积的奇偶性分析

GeoGebra 命令及参数

1. LaTeX(steps) → counter
2. Corner(1) → A
3. Corner(3) → B
4. Zip((2 * a), a, apowers) → a2powers
5. PrimeFactors(coeff1) → c1p
6. Unique(c1p) → c1pval
7. Frequency(c1p) → c1pcount
8. Zip(If[a / 2 ≟ floor(a / 2), a, a - 1], a, c1pcount) → c1peven
9. Zip(If[a / 2 ≟ floor(a / 2), 0, 1], a, c1pcount) → c1podd
10. If(Sum[c1podd] ≟ 0, "y", "n") → c1perfect
11. PrimeFactors(coeff2) → c2p
12. Unique(c2p) → c2pval
13. Frequency(c2p) → c2pcount
14. Zip(If[a / 2 ≟ floor(a / 2), a, a - 1], a, c2pcount) → c2peven
15. Zip(If[a / 2 ≟ floor(a / 2), 0, 1], a, c2pcount) → c2podd
16. If(Sum[c2podd] ≟ 0, "y", "n") → c2perfect
17. LaTeX((((((((("\sqrt{" * coeff1) * "a^{") * Element[a2powers, 1]) * "}}+\sqrt{") * coeff2) * "a^{") * Element[a2powers, 2]) * "}}")) → original
18. LaTeX("\text{Use the properties of radicals to rewrite the expression.}") → text1
19. LaTeX("\text{Rewrite each radical in the expression as a product of radicals, and evaluate}\cr\text{where possible.}") → text2
20. LaTeX("\text{Rewrite using the product property of radicals.}") → text5
21. LaTeX("\text{Evaluate the radicals of perfect squares.}") → text7
22. If(c1perfect ≟ "y", LaTeX[sqrt(coeff1)], LaTeX[(("\sqrt{" * coeff1) * "}")]) → c1evalperf
23. If(c2perfect ≟ "y", LaTeX[sqrt(coeff2)], LaTeX[(("\sqrt{" * coeff2) * "}")]) → c2evalperf
24. If(Element[apowers, 1] ≟ 1, LaTeX["a"], LaTeX[("a^" * Element[apowers, 1])]) → a1evalperf
25. If(Element[apowers, 2] ≟ 1, LaTeX["a"], LaTeX[("a^" * Element[apowers, 2])]) → a2evalperf
26. LaTeX(((((c1evalperf * a1evalperf) * " +") * c2evalperf) * a2evalperf)) → text8
27. LaTeX((((("\text{The radical expression }\textcolor{#000099}{" * original) * "}\text{ can be rewritten as }\textcolor{#000099}{") * text8) * "}.")) → text9
28. LaTeX("\text{Reduce any remaining radicals.}") → text10
29. LaTeX("\text{Factor out perfect squares within the radicand(s).}") → text13
30. Product(Zip[a^(b / 2), a, c1pval, b, c1peven]) → c1perf
31. Product(Zip[a^(b), a, c1pval, b, c1podd]) → c1rest
32. If(c1perfect ≟ "y", LaTeX[c1perf], If[c1perf ≟ 1, LaTeX[(("\sqrt{" * c1rest) * "}")], LaTeX[(((("\sqrt{" * c1perf^(2)) * "\cdot") * c1rest) * "}")]]) → writec1perf
33. Product(Zip[a^(b / 2), a, c2pval, b, c2peven]) → c2perf
34. Product(Zip[a^(b), a, c2pval, b, c2podd]) → c2rest
35. If(c2perfect ≟ "y", LaTeX[c2perf], If[c2perf ≟ 1, LaTeX[(("\sqrt{" * c2rest) * "}")], LaTeX[(((("\sqrt{" * c2perf^(2)) * "\cdot") * c2rest) * "}")]]) → writec2perf
36. LaTeX(((((writec1perf * a1evalperf) * " +") * writec2perf) * a2evalperf)) → text14
37. LaTeX("\text{Rewrite using the product property of radicals.}") → text15
38. If(c1perfect ≟ "y", LaTeX[c1perf], If[c1perf ≟ 1, LaTeX[(("\sqrt{" * c1rest) * "}")], LaTeX[(((("\sqrt{" * c1perf^(2)) * "}\cdot\sqrt{") * c1rest) * "}")]]) → writec1perfsplit
39. If(c2perfect ≟ "y", LaTeX[c2perf], If[c2perf ≟ 1, LaTeX[(("\sqrt{" * c2rest) * "}")], LaTeX[(((("\sqrt{" * c2perf^(2)) * "}\cdot\sqrt{") * c2rest) * "}")]]) → writec2perfsplit
40. LaTeX(((((writec1perfsplit * a1evalperf) * " +") * writec2perfsplit) * a2evalperf)) → text16
41. LaTeX("\text{Evaluate the radicals of perfect squares.}") → text17
42. If(c1perfect ≟ "y", LaTeX[c1perf], If[c1perf ≟ 1, LaTeX[(("\sqrt{" * c1rest) * "}")], LaTeX[(((c1perf * "\cdot\sqrt{") * c1rest) * "}")]]) → writec1perfroot
43. If(c2perfect ≟ "y", LaTeX[c2perf], If[c2perf ≟ 1, LaTeX[(("\sqrt{" * c2rest) * "}")], LaTeX[(((c2perf * "\cdot\sqrt{") * c2rest) * "}")]]) → writec2perfroot
44. LaTeX(((((writec1perfroot * a1evalperf) * " +") * writec2perfroot) * a2evalperf)) → text18
45. LaTeX((("\text{The remaining radical(s) cannot be further reduced. The final reduced expression}\cr\text{is }\textcolor{000099}{" * text18) * "}.")) → text19
46. LaTeX("\text{Determine whether the resulting expression is rational or irrational.}") → text20
47. LaTeX((((((((("\sqrt{" * coeff1) * "}\sqrt{a^{") * Element[a2powers, 1]) * "}}+\sqrt{") * coeff2) * "}\sqrt{a^{") * Element[a2powers, 2]) * "}}")) → text6
48. If(c1perfect ≟ "y" ∧ c2perfect ≟ "y", LaTeX[(("\text{The sum of two rational numbers is a rational number therefore }\cr\textcolor{#000099}{" * text18) * "}\text{ is rational.}")], LaTeX[(("\text{The sum of an irrational and either a rational or an irrational number is an}\cr\text{irrational number therefore }\textcolor{#000099}{" * text18) * "}\text{ is irrational.}")]) → text23
49. LaTeX("\text{Original expression}") → text3
50. LaTeX("\text{Move the slider to solve the problem.}") → text0
51. LaTeX((("\text{Reduce the radical expression }\textcolor{#000099}{" * original) * "}\text{. Assuming }\textcolor{#000099}\textit{a}\text{ is a whole number,}\cr\text{is the result rational or irrational?}")) → Problem
52. LaTeX("\text{Expression from the previous step}") → text11
53. If(c1perfect ≟ "y", LaTeX[((("\text{Because }\textcolor{#000099}\textit{a}\text{ is a whole number and }\textcolor{#000099}{" * writec1perfroot) * a1evalperf) * "}\text{ contains no radicals, the first part of the}\cr\text{expression is rational.}")], LaTeX[((("\text{Because }\textcolor{#000099}\textit{a}\text{ is a whole number and }\textcolor{#000099}{" * writec1perfroot) * a1evalperf) * "}\text{ contains an irreducible radical,}\cr\text{the first part of the expression is irrational.}")]) → text21
54. If(c2perfect ≟ "y", LaTeX[((("\text{Because }\textcolor{#000099}\textit{a}\text{ is a whole number and }\textcolor{#000099}{" * writec2perfroot) * a2evalperf) * "}\text{ contains no radicals, the second part of}\cr\text{the expression is rational.}")], LaTeX[((("\text{Because }\textcolor{#000099}\textit{a}\text{ is a whole number and }\textcolor{#000099}{" * writec2perfroot) * a2evalperf) * "}\text{ contains an irreducible radical,}\cr\text{the second part of the expression is irrational.}")]) → text22
55. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{to randomly generate a similar practice problem.}") → RandomizationText