查看“︁國中數學/國中數學七年級/2-5 指數律”︁的源代码

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|section=2-5 指數律
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在舊課綱中，該單元合併在[[國中數學/國中數學七年級/1-4 指數記法與科學記號|1-4 指數記法與科學記號]]。但在108新課綱後，該單元改為併入[[國中數學/國中數學七年級/2-3 分數的加減|2-3 分數的加減]]及[[國中數學/國中數學七年級/2-4 分數的乘除|2-4 分數的乘除]]。但因其重要性，因此將本單元獨立成一個單元。

==指數律==
<big><big><math>a,b,m,n,x</math>是不為<math>0</math>的實數(<math>a,b,m,n,x \in \mathbb{R} \neq 0</math>)，且a,b不可同時為負數</big></big>

===a<sup>m</sup> × a<sup>n</sup> = a<sup>m+n</sup>===
<math>
\begin{align}
& a^m \times a^n \\ 
= & (\underbrace{a \times a \times a \times \cdots \times a}_{m \text{個} \  a}) \times (\underbrace{a \times a \times \cdots \times a}_{n \text{個} \  a}) \\ 
= & \underbrace{a \times a \times a \times \cdots \times a}_{m \text{個} \  a} \times \underbrace{a \times a \times \cdots \times a}_{n \text{個} \  a} \\ 
= & \underbrace{a \times a \times a \times \cdots \times a}_{m+n \text{個} \  a} \\ 
= & a^{m+n}
\end{align}
</math>

===a<sup>m</sup> ÷ a<sup>n</sup>  = a<sup>m-n</sup>===
<math>
\begin{align}
& \text{(1)}\\
& \text{if} \quad m > n : \\ 
& a^m \div a^n \\ 
= & \frac{\overbrace{a \times a \times a \times \cdots \times a}^{m \text{個} \  a}}{\underbrace{a \times a \times \cdots \times a}_{n \text{個} \  a}} \\ 
= & \frac{\overbrace{\not{a} \times \not{a} \times \not{a} \times \not{\cdots} \times a \times a \times \cdots \times a}^{m \text{個} \  a}}{\underbrace{\not{a} \times \not{a} \times \not{\cdots} \times \not{a}}_{n \text{個} \  a}} \\ 
= & \underbrace{a \times a \times a \cdots a}_{m-n \text{個} \  a} \\ 
= & a^{m-n}\\
& \\
& \text{(2)}\\
& \text{if} \quad m < n : \\ 
& a^m \div a^n \\ 
= & \frac{\overbrace{a \times a \times \cdots \times a}^{m \text{個} \  a}}{\underbrace{a \times a \times a \times \cdots \times a}_{n \text{個} \  a}} \\ 
= & \frac{\overbrace{\not{a} \times \not{a} \times \not{\cdots} \times \not{a}}^{m \text{個} \  a}}{\underbrace{\not{a} \times \not{a} \times \not{a} \times \not{\cdots} \times a \times a \times \cdots \times a}_{n \text{個} \  a}} \\ 
= & \frac{1}{\underbrace{a \times a \times a \cdots a}_{n-m \text{個} \  a}} \\ 
= & \frac{1}{a^{n-m}}
\end{align}
</math>

===a<sup>0</sup> = 1===
<math>
\begin{align}
& \text{if} \quad m = n : \\ 
& a^m \div a^n \\ 
= & a^{m-n} = a^0 \\
= & \frac{\overbrace{a \times a \times a \times \cdots \times a}^{m \text{個} \  a}}{\underbrace{a \times a \times a \times \cdots \times a}_{n \text{個} \  a}} \\ 
= & \frac{\overbrace{\not{a} \times \not{a} \times \not{\cdots} \times \not{a}}^{m \text{個} \  a}}{\underbrace{\not{a} \times \not{a} \times \not{\cdots} \times \not{a}}_{n \text{個} \  a}} \\ 
= & 1 \\
\Rightarrow & a^0=1
\end{align}
</math>

===(a<sup>m</sup>)<sup>n</sup>   = a<sup>m×n</sup>===
<math>
\begin{align}
& (a^m)^n \\ 
= & (\underbrace{a^m \times a^m \times a^m \times \cdots \times a^m}_{n \text{個} \  a^m}) \\ 
= & \underbrace{{\overbrace{(a \times \cdots \times a)}^m} \times {\overbrace{(a \times \cdots \times a)}^m} \times \cdots \times {\overbrace{(a \times \cdots \times a)}^m}}_{m \times n \text{個} \  a} \\ 
= & \underbrace{a \times a \times a \times \cdots \times a}_{m \times n \text{個} \  a} \\ 
= & a^{m \times n}
\end{align}
</math>
===a<sup>m</sup> × b<sup>m</sup>  = (a × b)<sup>m</sup>===
<math>
\begin{align}
& {a^m} \times {b^m} \\ 
= & (\underbrace{a \times a \times \cdots \times a}_{m \text{個} \  a}) \times (\underbrace{b \times b \times \cdots \times b}_{m \text{個} \  b}) \\ 
= & {\underbrace{a \times a \times \cdots \times a}_{m \text{個} \  a}} \times {\underbrace{b \times b \times \cdots \times b}_{m \text{個} \  b}} \\ 
= & {\underbrace{a \times b \times a \times b \times a \times b \times a \times b \times \cdots \times a \times b}_{m \text{個} \  a \text{和} \  b \text{相 乘}}} \\ 
= & {\underbrace{(a \times b) \times (a \times b) \times (a \times b) \times (a \times b) \times \cdots \times (a \times b)}_{m \text{個} \  a \times \  b }} \\ 
= & (a \times b)^m
\end{align}
</math>
===a<sup>m</sup> ÷ b<sup>m</sup>  = (a ÷ b)<sup>m</sup>===
<math>
\begin{align}
& {a^m} \div {b^m} \\ 
= & (\underbrace{a \times a \times \cdots \times a}_{m \text{個} \  a}) \div (\underbrace{b \times b \times \cdots \times b}_{m \text{個} \  b}) \\ 
= & {\underbrace{a \times a \times \cdots \times a}_{m \text{個} \  a}} \div {\underbrace{b \div b \div \cdots \div b}_{m \text{個} \  b}} \\ 
= & {\underbrace{a \div b \times a \div b \times a \div b \times a \div b \times \cdots \times a \div b}_{m \text{個} \  a \text{和} \  b \text{相 除}}} \\ 
= & {\underbrace{(a \div b) \times (a \div b) \times (a \div b) \times (a \div b) \times \cdots \times (a \div b)}_{m \text{個} \  a \div \  b }} \\ 
= & (a \div b)^m
\end{align}
</math>
=== (<sup>b</sup>&frasl;<sub>a</sub>)<sup>m</sup> = <sup>b<sup>m</sup></sup>&frasl;<sub>a<sup>m</sup></sub> ===
<math>
\begin{align}
& (\frac{a}{b})^m \\ 
= & (a \div b)^m \\
= & {a^m} \div {b^m} \\
= & \frac{a^m}{b^m}
\end{align}
</math>

=== a<sup>-x</sup> = <sup>1</sup>&frasl;<sub>a<sup>x</sup></sub>===
<math>
\begin{align}
& a^{-x} \\ 
= & (a^{-1})^x \\ 
= & (a^{m-n})^x \quad (m-n=-1) \\ 
= & (a^m - a^n)^x \\ 
= & \bigg(\frac{\overbrace{a \times a \times \cdots \times a}^{m \text{個} \  a}}{\underbrace{a \times a \times a \times \cdots \times a}_{n \text{個} \  a}}\bigg)^x \\ 
= & \frac{(\overbrace{a \times a \times \cdots \times a}^{m \text{個} \  a})^x}{(\underbrace{a \times a \times a \times \cdots \times a}_{n \text{個} \  a})^x} \\ 
= & \frac{\overbrace{{a^x} \times {a^x} \times \cdots \times {a^x}}^{m \text{個} \  {a^x}}}{\underbrace{{a^x} \times {a^x} \times {a^x} \times \cdots \times {a^x}}_{n \text{個} \  {a^x}}} \\ 
= & \frac{\overbrace{\not{a^x} \times \not{a^x} \times \not{\cdots} \times \not{a^x}}^{m \text{個} \  a}}{\underbrace{\not{a^x} \times \not{a^x} \times \not{a^x} \times \not{\cdots} \times {a^x}}_{n \text{個} \  {a^x}}} \\ 
= & \frac{1}{\underbrace{a^x}_{n-m=1 \  \text{個} \  {a^x}}} \\ 
= & \frac{1}{a^x}
\end{align}
</math>

==習題==
1.<math>(10 \times {5^2}) \div (2 \times 5^{-3}) = ?</math><br>
(A) 1 (B) 5<sup>4</sup> (C) 5<sup>5</sup> (D) 5<sup>6</sup> <br>
2.<math>(-4^4)^2 \div 2^{10}\div 8^2</math><br>
(A) 1 (B) 2 (C) 4 (D) 8

==習題解答及解析==
1.(D) 2.(A)<br>
1.<br>
<math>
\begin{align}
& (10 \times {5^2}) \div (2 \times 5^{-3}) \\
= & 10 \times {5^2} \div 2 \div 5^{-3} \\
= & (10 \div 2) \times ({5^2} \div 5^{-3}) \\
= & 5 \times 5^{2-(-3)} \\
= & 5^1 \times 5^5 \\
= & 5^6
\end{align}
</math><br>
2.<br>
<math>
\begin{align}
& (-4^4)^2 \div 2^{10} \div 8^2 \\
= & 2^{16} \div 2^{10} \div 2^6 \\
= & 2^{(16-10-6)} \\
= & 2^0 \\
= & 1
\end{align}
</math><br>
[[Category:國中數學]]