5.2: 指数乘积规则
- Page ID
- 171084
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)对于任何实数\(a\)和正数\(m\)\(n\)，指数的乘积规则如下。
\(a^m \cdot a^n = a^{m+n}\)
注意：基础必须相同才能使用乘积规则。
想法：
从最后一节开始\(x^3 = \textcolor{blue}{ x \cdot x \cdot x }\qquad x^5 = \textcolor{red}{x \cdot x \cdot x \cdot x \cdot x}\)
他们的产品
\(x^3 \cdot x^5 = \textcolor{blue}{x \cdot x \cdot x} \textcolor{red}{\cdot x \cdot x \cdot x \cdot x \cdot x} = x^8\)
因此，\(x^3 \cdot x^5 = x^{3+5 }= x^8\)
使用指数乘积法则来简化表达式。
- \(k^3 \cdot k^9\)
- \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\)
- \((−2a)^3 \cdot (−2a)^7\)
- \(x \cdot x^3 \cdot x^{11}\)
- \(y^{13 }\cdot y^{33}\)
- \(x^3 \cdot y^2 \cdot x \cdot y^4\)
解决方案
| 表情 | 产品规则 | 基地 |
| \(k^3 \cdot k^9\) | \(k^{3+9}= k^{12}\) | \(k\) |
| \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\) | \(\left( \dfrac{2 }{7}\right)^{2+6 }= \left(\dfrac{2 }{7}\right)^8\) | \(\dfrac{2}{7}\) |
| \((−2a)^3 \cdot (−2a)^7\) | \((−2a)^{3+7 }= (−2a)^{10}\) | \(-2a\) |
| \(x \cdot x^3 \cdot x^{11}\) | \(x ^{1+3+11 }= x^{15}\) | \(x\) |
| \(y^{13 }\cdot y^{33}\) | \(y^{13+33 }= y^46\) | \(y\) |
| \(x^3 \cdot y^2 \cdot x \cdot y^4\) | \(x^{3+1 }\cdot y ^{2+4 }= x^{ 4 }\cdot y^{6}\) | \(x\)和\(y\) |
注意：同样，基数必须相同，以便使用指数乘积法则进行简化
简化使用指数乘积法则的有用步骤：
- 识别具有共同基础的术语
- 确定常用基数的指数。
- 将常用基数的指数相加，并使总和的结果成为新的指数。
- 根据需要重复步骤
使用指数乘积法则简化以下内容。
- \(f^3 \cdot f^11\)
- \(\left(\dfrac{x}{7}\right)^2 \cdot \left(\dfrac{x }{7}\right)^3\)
- \((−7x)^9 \cdot (−7x)^7\)
- \(h^5 \cdot h^3 \cdot h^{11}\)
- \(t^{13} \cdot t^{33}\)
- \(x^8 \cdot y^2 \cdot z \cdot x^ 3 \cdot y^2 \cdot z^{17}\)
- \(x^3 \cdot y^4 \cdot x^3\)