GeoGebra Algebra & Real Numbers Example

GeoGebra Commands & Parameters

1. GCD(n, d) → greatestfact
2. PrimeFactors(greatestfact) → greatfactp
3. Length(greatfactp) → lengreatfactp
4. If(lengreatfactp ≟ 0, {LaTeX["\text{}1"]}, If[lengreatfactp ≟ 1, {LaTeX[("\text{}" * Element[greatfactp, 1])]}, If[lengreatfactp ≟ 2, {LaTeX[("\text{}" * Element[greatfactp, 1])], LaTeX[("\text{ and }" * Element[greatfactp, Length[greatfactp]])]}, Append[Append[LaTeX[("\text{}" * Element[greatfactp, 1])], Sequence[LaTeX[("\text{, }" * Element[greatfactp, i])], i, 2, lengreatfactp - 1]], LaTeX[("\text{, and }" * Element[greatfactp, lengreatfactp])]]]]) → greatfactplong
5. PrimeFactors(dred) → dredp
6. Unique(dredp) → dredunique
7. Frequency(dredp) → dredpcount
8. Zip(floor(a / 2), a, dredpcount) → dhalfpow
9. Zip(a^(b), a, dredunique, b, dhalfpow) → dnonroots
10. Product(dnonroots) → dnonrootprod
11. Zip(If[(2 * a) ≟ b, 0, 1], a, dhalfpow, b, dredpcount) → dextras
12. Zip(a^(b), a, dredunique, b, dextras) → droots
13. Product(droots) → drootprod
14. PrimeFactors(nred) → nredp
15. Unique(nredp) → nredunique
16. Frequency(nredp) → nredpcount
17. Zip(floor(a / 2), a, nredpcount) → nhalfpow
18. Zip(a^(b), a, nredunique, b, nhalfpow) → nnonroots
19. Product(nnonroots) → nnonrootprod
20. Zip(If[(2 * a) ≟ b, 0, 1], a, nhalfpow, b, nredpcount) → nextras
21. Zip(a^(b), a, nredunique, b, nextras) → nroots
22. Product(nroots) → nrootprod
23. PrimeFactors(d) → dp
24. Length(dp) → dfull
25. PrimeFactors(n) → np
26. Length(np) → nfull
27. LaTeX(steps) → counter
28. Corner(1) → A
29. Corner(3) → B
30. Unique(dp) → dpunique
31. Frequency(dp) → dpcount
32. Unique(np) → npunique
33. Frequency(np) → npcount
34. Length(npcount) → nplen
35. Length(dpcount) → dplen
36. Join({{If[Element[npcount, 1] ≟ 1, LaTeX[Element[npunique, 1]], LaTeX[((Element[npunique, 1] * "^") * Element[npcount, 1])]]}, Sequence[If[Element[npcount, i] ≟ 1, LaTeX[("\cdot" * Element[npunique, i])], LaTeX[((("\cdot" * Element[npunique, i]) * "^") * Element[npcount, i])]], i, 2, nplen]}) → npg
37. Join({{If[Element[dpcount, 1] ≟ 1, LaTeX[Element[dpunique, 1]], LaTeX[((Element[dpunique, 1] * "^") * Element[dpcount, 1])]]}, Sequence[If[Element[dpcount, i] ≟ 1, LaTeX[("\cdot" * Element[dpunique, i])], LaTeX[((("\cdot" * Element[dpunique, i]) * "^") * Element[dpcount, 1])]], i, 2, dplen]}) → dpg
38. Join({{LaTeX[Element[np, 1]]}, Sequence[LaTeX[("\cdot" * Element[np, i])], i, 2, nfull]}) → nplong
39. Join({{LaTeX[Element[dp, 1]]}, Sequence[LaTeX[("\cdot" * Element[dp, i])], i, 2, dfull]}) → dplong
40. If((nnonrootprod ≠ 1) ∧ (nrootprod ≠ 1), LaTeX[(((nnonrootprod * "\sqrt{") * nrootprod) * "}")], If[nnonrootprod ≠ 1, LaTeX[nnonrootprod], If[nrootprod ≠ 1, LaTeX[(("\sqrt{" * nrootprod) * "}")], LaTeX["1"]]]) → npretty2
41. If((dnonrootprod ≠ 1) ∧ (drootprod ≠ 1), LaTeX[(((dnonrootprod * "\sqrt{") * drootprod) * "}")], If[dnonrootprod ≠ 1, LaTeX[dnonrootprod], If[drootprod ≠ 1, LaTeX[(("\sqrt{" * drootprod) * "}")], LaTeX["1"]]]) → dpretty2
42. If(dform ≟ 1 ∧ nform ≟ 0, LaTeX[(((("\sqrt{\frac{" * n) * "}{") * dppretty) * "}}")], If[dform ≟ 0 ∧ nform ≟ 1, LaTeX[(((("\sqrt{\frac{" * nppretty) * "}{") * d) * "}}")], If[dform ≟ 0 ∧ nform ≟ 0, LaTeX[(((("\sqrt{\frac{" * n) * "}{") * d) * "}}")], LaTeX["error"]]]) → nugget
43. LaTeX("\text{Rewrite each number in the expression as a product of prime numbers. Then group}\cr\text{identical factors together using exponents.}") → text1
44. LaTeX(nugget) → text3
45. LaTeX((((("\text{Rewrite " * n) * " and ") * d) * " as a products of}\cr\text{their prime factors.}")) → text4c
46. LaTeX((("\text{Rewrite " * d) * " as a product of its prime}\cr\text{factors.}")) → text4b
47. LaTeX((("\text{Rewrite " * n) * " as a product of its prime}\cr\text{factors.}")) → text4a
48. LaTeX((((("\sqrt{\frac{" * nppretty) * "}{") * dppretty) * "}}")) → text7
49. LaTeX("\text{Group prime factors together using exponents.}") → text6
50. LaTeX("\text{Divide out factors that appear in both the numerator and the denominator.}") → text8
51. Length(greatfactp) → gfplen
52. Length(nredpcount) → nredplen
53. Length(dredpcount) → dredplen
54. Join({{If[Element[dredpcount, 1] ≟ 1, LaTeX[Element[dredunique, 1]], LaTeX[((Element[dredunique, 1] * "^") * Element[dredpcount, 1])]]}, Sequence[If[Element[dredpcount, i] ≟ 1, LaTeX[("\cdot" * Element[dredunique, i])], LaTeX[((("\cdot" * Element[dredunique, i]) * "^") * Element[dredpcount, i])]], i, 2, dredplen]}) → dredpglong
55. Join({{If[Element[nredpcount, 1] ≟ 1, LaTeX[Element[nredunique, 1]], LaTeX[((Element[nredunique, 1] * "^") * Element[nredpcount, 1])]]}, Sequence[If[Element[nredpcount, i] ≟ 1, LaTeX[("\cdot" * Element[nredunique, i])], LaTeX[((("\cdot" * Element[nredunique, i]) * "^") * Element[nredpcount, i])]], i, 2, nredplen]}) → nredpglong
56. LaTeX((((("\sqrt{\frac{" * nredpretty) * "}{") * dredpretty) * "}}")) → text12
57. LaTeX((("\text{The numerator and denominator have }" * greatfactpretty) * "\cr\text{as common factor(s). Cancel the common factor(s).}")) → text11
58. LaTeX("\text{Use the properties of radicals to rewrite the reduced expression.}") → text13
59. LaTeX((((("\frac{\sqrt{" * nredpretty) * "}}{\sqrt{") * dredpretty) * "}}")) → text17
60. LaTeX("\text{Rewrite using the quotient property of radicals.}") → text16
61. Zip(a - 1, a, nredpcount) → nredpcountdown
62. Zip(a - 1, a, dredpcount) → dredpcountdown
63. Zip(If[b ≟ 1, LaTeX[(("\sqrt{" * a) * "}")], If[(b > 1) ∧ (b / 2 ≠ floor(b / 2)), LaTeX[(((((("\sqrt{" * a) * "^") * c) * "}\sqrt{") * a) * "}")], LaTeX[(((("\sqrt{" * a) * "^") * b) * "}")]]], a, nredunique, b, nredpcount, c, nredpcountdown) → nredsplit
64. Zip(If[b ≟ 1, LaTeX[(("\sqrt{" * a) * "}")], If[(b > 1) ∧ (b / 2 ≠ floor(b / 2)), LaTeX[(((((("\sqrt{" * a) * "^") * c) * "}\sqrt{") * a) * "}")], LaTeX[(((("\sqrt{" * a) * "^") * b) * "}")]]], a, dredunique, b, dredpcount, c, dredpcountdown) → dredsplit
65. LaTeX((((("\frac{" * nredsplitpretty) * "}{") * dredsplitpretty) * "}")) → text19
66. Zip(If[a / 2 ≟ floor(a / 2), a / 2, (a + 1) / 2], a, nredpcountdown) → nredpcounthalf
67. Zip(If[a / 2 ≟ floor(a / 2), a / 2, (a + 1) / 2], a, dredpcountdown) → dredpcounthalf
68. Zip(If[b ≟ 1, LaTeX[(("\sqrt{" * a) * "}")], If[(b > 1) ∧ (b / 2 ≠ floor(b / 2)), If[c > 1, LaTeX[(((((a * "^") * c) * "\sqrt{") * a) * "}")], LaTeX[(((a * "\sqrt{") * a) * "}")]], If[c > 1, LaTeX[((a * "^") * c)], LaTeX[a]]]], a, nredunique, b, nredpcount, c, nredpcounthalf) → nredsimp
69. Zip(If[b ≟ 1, LaTeX[(("\sqrt{" * a) * "}")], If[(b > 1) ∧ (b / 2 ≠ floor(b / 2)), If[c > 1, LaTeX[(((((a * "^") * c) * "\sqrt{") * a) * "}")], LaTeX[(((a * "\sqrt{") * a) * "}")]], If[c > 1, LaTeX[((a * "^") * c)], LaTeX[a]]]], a, dredunique, b, dredpcount, c, dredpcounthalf) → dredsimp
70. LaTeX((((("\frac{" * nredsimppretty) * "}{") * dredsimppretty) * "}")) → text21
71. If(drootprod ≟ 1 ∧ dnonrootprod ≟ 1, LaTeX[npretty2], LaTeX[(((("\frac{" * npretty2) * "}{") * dpretty2) * "}")]) → text23
72. LaTeX(text23) → text26
73. LaTeX("\text{Multiply the numerator and the denominator by the}\cr\text{radical in the denominator.}") → text27
74. LaTeX((((((text23 * "\cdot\frac{\sqrt{") * drootprod) * "}}{\sqrt{") * drootprod) * "}}")) → text28
75. If((nnonrootprod ≠ 1) ∧ (nfrootprod ≠ 1), LaTeX[(((nnonrootprod * "\sqrt{") * nfrootprod) * "}")], If[nnonrootprod ≠ 1, LaTeX[nnonrootprod], If[nfrootprod ≠ 1, LaTeX[(("\sqrt{" * nfrootprod) * "}")], LaTeX["1"]]]) → nfinal
76. If(dfnonrootprod ≟ 1, LaTeX[nfinal], LaTeX[(((("\frac{" * nfinal) * "}{") * dfnonrootprod) * "}")]) → text30
77. LaTeX((((("\text{The expression }" * nugget) * "\text{ is equal to }") * text30) * ".")) → text31
78. LaTeX("\text{Rationalize the denominator if there is a square root in it.}") → text24
79. LaTeX("\text{There isn't a square root in the denominator so nothing needs to be done.}") → text27alt
80. LaTeX("\text{Please skip to step 31.}") → text28alt
81. LaTeX("\text{Determine whether the resulting expression is rational or irrational.}") → text32
82. LaTeX("\text{Since the final expression contains a square root, the expression is irrational.}") → text35a
83. LaTeX("\text{Since the final expression is a ratio of integers, the expression is rational.}") → text35b
84. LaTeX((((("\sqrt{\frac{" * nppretty) * "}{") * dpretty) * "}}")) → text5b
85. LaTeX("\text{Factor out perfect squares and then rewrite using the}\cr\text{product propert of radicals.}") → text18
86. LaTeX((("\text{Reduce the radical expression }" * nugget) * "\text{. If the result has a root in the denominator,}\cr\text{rationalize the denominator. Is the final expression rational or irrational?}")) → Problem
87. LaTeX("\text{Evaluate the radicals of perfect squares.}") → text20
88. LaTeX("\text{Evaluate any powers and products.}") → text22
89. LaTeX("\text{Multiply any radicals in the numerator and the}\cr\text{radicals in the denominator.}") → text29
90. LaTeX((((("\sqrt{\frac{" * npretty) * "}{") * dppretty) * "}}")) → text5a
91. LaTeX((((("\sqrt{\frac{" * npretty) * "}{") * dpretty) * "}}")) → text5c
92. LaTeX("\text{Move the slider to solve the problem.}") → text0
93. LaTeX("\text{Original expression}") → text2
94. LaTeX("\text{Expression from previous step}") → text9
95. LaTeX("\text{Expression from previous step}") → text14
96. LaTeX("\text{Expression from previous step}") → text25
97. LaTeX("\text{Expression from previous step}") → text33
98. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{to randomly generate a similar practice problem.}") → RandomizationText