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简化含负指数的代数表达式

简化含负指数的代数表达式
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2015-08-26 20:44
解析几何几何作图
①本范例演示如何简化包含负指数的代数表达式，并转换为正指数形式；②涉及的核心数学概念为指数运算法则，特别是同底数幂的乘法与指数的加减规则；③作图要点提炼自构造摘要，用到了 LaTeX 命令进行数学公式显示，以及 Border, Gear, Book, Nexbl, prebl, nexgr, pregr, Probar, Stepmax, Slider, Stepof, P, Q, DatakitBG, Dataclose, WordstoknowBG, Wordsclose, Originalbutton, RandomButton, Quadratic, Linear, Constant, Refresh_1, Problue1 至 Problue17 等对象进行界面布局与交互设计；④适合高中阶段学习指数运算的学生，可用于课堂演示或课后练习。

GeoGebra 命令及参数

  1. LaTeX("\text{Use the exponent rule \textcolor{#000099}{a^m\; • \;a^n = a^{m +n}} to simplify the expression.}") → Pg1B
  2. LaTeX("\text{To multiply powers with the same base, add the exponents.}") → Pg1C
  3. LaTeX("\text{Use the exponent rule \textcolor{#000099}{a^0 = 1} to further simplify the expression.}") → Pg1G
  4. LaTeX("\text{Use the exponent rule \textcolor{#000099}{a^{−m} = \frac{1}{a^m}} to write the negative exponent as a positive exponent.}") → Pg2A
  5. LaTeX("\text{To rewrite a negative exponent as a positive one, take its reciprocal.}") → Pg2B
  6. LaTeX("\text{\textcolor{#000099}{a^{−m} = \frac{1}{a^m}}}") → Pg2C
  7. LaTeX("\text{\mathbf{Exponent} \text the number of times a factor is being multiplied together in an}") → WTK1
  8. LaTeX((((((("\text{Simplify the expression \textcolor{#000099}{x^–^" * X1P) * "\; •\; y^0\; • \; z^–^") * X2P) * "\;•\; x^") * YP) * "}. Write the expression using only positive exponents.}")) → Pg1A
  9. LaTeX((((((((((("\text{In the expression \textcolor{#000099}{x^–^" * X1P) * "\; •\; y^0\; • \; z^–^") * X2P) * "\;•\; x^") * YP) * "}, the only terms with the same base are \textcolor{#000099}{x^-^") * X1P) * "} and \textcolor{#000099}{x^") * YP) * "}. To multiply these terms, add the exponents.}")) → Pg1D
  10. LaTeX((((((((((("\text{\textcolor{#000099}{x^-^" * X1P) * " \;•\; x^") * YP) * " = x^-^{") * X1P) * " + ") * YP) * "} = x^{") * X12P) * "}}}")) → Pg1E
  11. LaTeX((((((((((("\text{The expression \textcolor{#000099}{x^–^" * X1P) * "\; •\; y^0\; • \; z^–^") * X2P) * "\;•\; x^") * YP) * "} can now be simplified to \textcolor{#000099}{x^{") * X12P) * "} • \;y^0 •\; z^-^{") * X2P) * "}.}}")) → Pg1F
  12. LaTeX("\text{Any value to the \textcolor{#000099}{0} power equals \textcolor{#000099}{1}. Thus, \textcolor{#000099}{y^0 = 1}.}") → Pg1H
  13. LaTeX("\text{Substitute \textcolor{#000099}{1} for \textcolor{#000099}{y^0}.}") → Pg1I
  14. LaTeX((((("\text{The expression is now simplified to \textcolor{#000099}{x^{" * X12P) * "} • \;z^{–") * X2P) * "}.}}")) → Pg1K
  15. LaTeX((((("\text{\textcolor{#000099}{z^{−" * X2P) * "} = \frac{1}{z^") * X2P) * "}}}")) → Pg2D
  16. LaTeX((((("\text{Substitute \textcolor{#000099}{\frac{1}{z^" * X2P) * "}} for \textcolor{#000099}{ z^{–") * X2P) * "}}.}")) → Pg2E
  17. LaTeX((((((((((("\text{The expression \textcolor{#000099}{x^–^" * X1P) * "\; •\; y^0\; • \; z^–^") * X2P) * "\;•\; x^") * YP) * "} can be simplified to \textcolor{#000099}{\frac{x^{") * X12P) * "}}{z^") * X2P) * "}}.}")) → Pg2G
  18. LaTeX((((((((((((("\text{\textcolor{#000099}{x^{" * X12P) * "} •\; z^{−") * X2P) * "} = x^{") * X12P) * "} • \frac{1}{z^") * X2P) * "} = \frac{x^{") * X12P) * "}}{z^") * X2P) * "} }}")) → Pg2F
  19. LaTeX((("\text{Exponent of z = " * Constant) * "}")) → DK2
  20. LaTeX("\text{exponential expression; in the expression \textcolor{#000099}{a^b},\textcolor{#000099}{b} is the exponent.}") → Wtk2
  21. LaTeX((((((((((((("\text{\textcolor{#000099}{x^{" * X12P) * "} • \;y^0 •\; z^{–") * X2P) * "} = x^{") * X12P) * "} • \;1\;•\;z^{–") * X2P) * "} \; = x^{") * X12P) * "} • \;z^{–") * X2P) * "}}}")) → Pg1J
  22. LaTeX((("\text{Exponent of X₁ = " * Quadratic) * "}")) → DK1
  23. LaTeX((("\text{Exponent of X₂ = " * Linear) * "}")) → DK3