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Simplifying Negative Exponent Expressions

Simplifying Negative Exponent Expressions
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2015-08-26 20:44
Coordinate GeometryGeometric Construction
This example demonstrates how to simplify algebraic expressions containing negative exponents and rewrite them using only positive exponents. The core math concepts involved are exponent rules, particularly the multiplication of like bases and the addition/subtraction of exponents. The construction leverages the LaTeX command for mathematical formula display and various GeoGebra objects such as Border, Gear, Book, Nexbl, prebl, nexgr, pregr, Probar, Stepmax, Slider, Stepof, P, Q, DatakitBG, Dataclose, WordstoknowBG, Wordsclose, Originalbutton, RandomButton, Quadratic, Linear, Constant, Refresh_1, and Problue1-17 for interface layout and interactive design. It is suitable for high school students learning exponent operations, serving as a classroom demonstration or homework exercise.

GeoGebra Commands & Parameters

  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