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简化表达式并使用正指数

简化表达式并使用正指数
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2015-09-11 21:52
函数解析几何
所属专题 CCSS Support Algebra I Unit 1
①本范例演示如何简化包含负指数和零指数的代数表达式，并转换为正指数形式；②涉及的核心数学概念包括指数法则、同底数幂相乘、任何数的零次幂等于1；③作图要点使用了GeoGebra的LaTeX命令进行数学公式显示，未涉及具体作图命令；④适合初中及以上学生学习指数运算规则，为后续微积分和高等数学打下基础。

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{\textcolor{#000099}{x^" * X1P) * " \;•\; x^") * X2P) * " = x^{") * X1P) * " + ") * X2P) * "} = x^{") * X12P) * "}}}")) → Pg1E
  3. LaTeX("\text{Use the exponent rule \textcolor{#000099}{a^0 = 1} to further simplify the expression.}") → Pg1G
  4. LaTeX("\text{Substitute 1 for \textcolor{#000099}{z^0}.}") → Pg1I
  5. LaTeX((((((((((((("\text{\textcolor{#000099}{x^{" * X12P) * "} • \;y^{–") * YP) * "} •\; z^0 = x^{") * X12P) * "} • \;y^{–") * YP) * "} •\; 1 = x^{") * X12P) * "} • \;y^{–") * YP) * "}}}")) → Pg1J
  6. LaTeX("\text{Use the exponent rule \textcolor{#000099}{a^{−m} = \frac{1}{a^m}} to write the negative exponent as a positive exponent.}") → Pg2A
  7. LaTeX("\text{\textcolor{#000099}{a^{−m} = \frac{1}{a^m}}}") → Pg2C
  8. LaTeX((((("\text{\textcolor{#000099}{y^{−" * YP) * "} = \frac{1}{y^") * YP) * "}}}")) → Pg2D
  9. LaTeX((((("\text{Substitute \textcolor{#000099}{\frac{1}{y^" * YP) * "}} for \textcolor{#000099}{ y^{–") * YP) * "}}.}")) → Pg2E
  10. LaTeX((((((((((((("\text{\textcolor{#000099}{x^{" * X12P) * "} •\; y^{−") * YP) * "} = x^{") * X12P) * "} • \frac{1}{y^") * YP) * "} = \frac{x^{") * X12P) * "}}{y^") * YP) * "} }}")) → Pg2F
  11. LaTeX((((((((((("\text{The expression \textcolor{#000099}{x^" * X1P) * " •\; y^–^") * YP) * "\; • \; x^") * X2P) * "•\; z^0} can be simplified to \textcolor{#000099}{\frac{x^{") * X12P) * "}}{y^") * YP) * "}}.}")) → Pg2G
  12. LaTeX((((((((((("\text{The expression \textcolor{#000099}{x^" * X1P) * " \;•\; y^–^") * YP) * "\; • \; x^") * X2P) * "•\; z^0} can now be simplified to \textcolor{#000099}{x^{") * X12P) * "} • \;y^{–") * YP) * "} •\; z^0}.}")) → Pg1F
  13. LaTeX((((("\text{The expression is now simplified to \textcolor{#000099}{x^{" * X12P) * "} •\; y^{–") * YP) * "}}.}")) → Pg1K
  14. LaTeX((("\text{Exponent of y = -" * Linear) * "}")) → DK3
  15. LaTeX((("\text{Exponent of x₁ = " * Quadratic) * "}")) → DK1
  16. LaTeX((("\text{Exponent of x₂ = " * Constant) * "}")) → DK2
  17. LaTeX((((((("\text{Simplify the expression \textcolor{#000099}{x^" * X1P) * "\; •\; y^–^") * YP) * "\; • \; x^") * X2P) * "\;•\; z^0}. Write the expression using only positive exponents.}")) → Pg1A
  18. LaTeX("\text{To multiply powers with the same base, add the exponents.}") → Pg1C
  19. LaTeX((((((((((("\text{In the expression \textcolor{#000099}{x^" * X1P) * " \;•\; y^–^") * YP) * "\; • \; x^") * X2P) * "\;•\; z^0}, the only terms with the same base are \textcolor{#000099}{x^") * X1P) * "} and \textcolor{#000099}{x^") * X2P) * "}. To multiply these terms, add the exponents.}")) → Pg1D
  20. LaTeX("\text{Any value to the 0 power equals 1. Thus, \textcolor{#000099}{z^0 = 1}.}") → Pg1H
  21. LaTeX("\text{To rewrite a negative exponent as a positive one, take its reciprocal.}") → Pg2B
  22. LaTeX("\text{\mathbf{exponent} \text the number of times a factor is being multiplied together in an}") → WTK1
  23. LaTeX("\text{exponential expression; in the expression \textcolor{#000099}{a^b}, \textcolor{#000099}{b} is the exponent}") → Wtk2