国中数学/国中数学七年级/2-5 指数律

2-4 分数的乘除

◄

国中数学七年级 2-5 指数律

►

3-1 一元一次式

在旧课纲中，该单元合并在1-4 指数记法与科学记号。但在108新课纲后，该单元改为并入2-3 分数的加减及2-4 分数的乘除。但因其重要性，因此将本单元独立成一个单元。

指数律

$a,b,m,n,x$是不为$0$的实数($a,b,m,n,x\in \mathbb {R} \neq 0$)，且a,b不可同时为负数

a^m × aⁿ = a^m+n

${\begin{aligned}&a^{m}\times a^{n}\\=&(\underbrace {a\times a\times a\times \cdots \times a} _{m{\text{個}}\ a})\times (\underbrace {a\times a\times \cdots \times a} _{n{\text{個}}\ a})\\=&\underbrace {a\times a\times a\times \cdots \times a} _{m{\text{個}}\ a}\times \underbrace {a\times a\times \cdots \times a} _{n{\text{個}}\ a}\\=&\underbrace {a\times a\times a\times \cdots \times a} _{m+n{\text{個}}\ a}\\=&a^{m+n}\end{aligned}}$

a^m ÷ aⁿ = a^m-n

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

a⁰ = 1

${\begin{aligned}&{\text{if}}\quad m=n:\\&a^{m}\div a^{n}\\=&a^{m-n}=a^{0}\\=&{\frac {\overbrace {a\times a\times a\times \cdots \times a} ^{m{\text{個}}\ a}}{\underbrace {a\times a\times a\times \cdots \times a} _{n{\text{個}}\ a}}}\\=&{\frac {\overbrace {\not {a}\times \not {a}\times \not {\cdots }\times \not {a}} ^{m{\text{個}}\ a}}{\underbrace {\not {a}\times \not {a}\times \not {\cdots }\times \not {a}} _{n{\text{個}}\ a}}}\\=&1\\\Rightarrow &a^{0}=1\end{aligned}}$

(a^m)ⁿ = a^m×n

${\begin{aligned}&(a^{m})^{n}\\=&(\underbrace {a^{m}\times a^{m}\times a^{m}\times \cdots \times a^{m}} _{n{\text{個}}\ a^{m}})\\=&\underbrace {{\overbrace {(a\times \cdots \times a)} ^{m}}\times {\overbrace {(a\times \cdots \times a)} ^{m}}\times \cdots \times {\overbrace {(a\times \cdots \times a)} ^{m}}} _{m\times n{\text{個}}\ a}\\=&\underbrace {a\times a\times a\times \cdots \times a} _{m\times n{\text{個}}\ a}\\=&a^{m\times n}\end{aligned}}$

a^m × b^m = (a × b)^m

${\begin{aligned}&{a^{m}}\times {b^{m}}\\=&(\underbrace {a\times a\times \cdots \times a} _{m{\text{個}}\ a})\times (\underbrace {b\times b\times \cdots \times b} _{m{\text{個}}\ b})\\=&{\underbrace {a\times a\times \cdots \times a} _{m{\text{個}}\ a}}\times {\underbrace {b\times b\times \cdots \times b} _{m{\text{個}}\ b}}\\=&{\underbrace {a\times b\times a\times b\times a\times b\times a\times b\times \cdots \times a\times b} _{m{\text{個}}\ a{\text{和}}\ b{\text{相 乘}}}}\\=&{\underbrace {(a\times b)\times (a\times b)\times (a\times b)\times (a\times b)\times \cdots \times (a\times b)} _{m{\text{個}}\ a\times \ b}}\\=&(a\times b)^{m}\end{aligned}}$

a^m ÷ b^m = (a ÷ b)^m

${\begin{aligned}&{a^{m}}\div {b^{m}}\\=&(\underbrace {a\times a\times \cdots \times a} _{m{\text{個}}\ a})\div (\underbrace {b\times b\times \cdots \times b} _{m{\text{個}}\ b})\\=&{\underbrace {a\times a\times \cdots \times a} _{m{\text{個}}\ a}}\div {\underbrace {b\div b\div \cdots \div b} _{m{\text{個}}\ b}}\\=&{\underbrace {a\div b\times a\div b\times a\div b\times a\div b\times \cdots \times a\div b} _{m{\text{個}}\ a{\text{和}}\ b{\text{相 除}}}}\\=&{\underbrace {(a\div b)\times (a\div b)\times (a\div b)\times (a\div b)\times \cdots \times (a\div b)} _{m{\text{個}}\ a\div \ b}}\\=&(a\div b)^{m}\end{aligned}}$

(^b⁄_a)^m = ^bm⁄_a^m

${\begin{aligned}&({\frac {a}{b}})^{m}\\=&(a\div b)^{m}\\=&{a^{m}}\div {b^{m}}\\=&{\frac {a^{m}}{b^{m}}}\end{aligned}}$

a^-x = ¹⁄_a^x

${\begin{aligned}&a^{-x}\\=&(a^{-1})^{x}\\=&(a^{m-n})^{x}\quad (m-n=-1)\\=&(a^{m}-a^{n})^{x}\\=&{\bigg (}{\frac {\overbrace {a\times a\times \cdots \times a} ^{m{\text{個}}\ a}}{\underbrace {a\times a\times a\times \cdots \times a} _{n{\text{個}}\ a}}}{\bigg )}^{x}\\=&{\frac {(\overbrace {a\times a\times \cdots \times a} ^{m{\text{個}}\ a})^{x}}{(\underbrace {a\times a\times a\times \cdots \times a} _{n{\text{個}}\ a})^{x}}}\\=&{\frac {\overbrace {{a^{x}}\times {a^{x}}\times \cdots \times {a^{x}}} ^{m{\text{個}}\ {a^{x}}}}{\underbrace {{a^{x}}\times {a^{x}}\times {a^{x}}\times \cdots \times {a^{x}}} _{n{\text{個}}\ {a^{x}}}}}\\=&{\frac {\overbrace {\not {a^{x}}\times \not {a^{x}}\times \not {\cdots }\times \not {a^{x}}} ^{m{\text{個}}\ a}}{\underbrace {\not {a^{x}}\times \not {a^{x}}\times \not {a^{x}}\times \not {\cdots }\times {a^{x}}} _{n{\text{個}}\ {a^{x}}}}}\\=&{\frac {1}{\underbrace {a^{x}} _{n-m=1\ {\text{個}}\ {a^{x}}}}}\\=&{\frac {1}{a^{x}}}\end{aligned}}$

习题

1.$(10\times {5^{2}})\div (2\times 5^{-3})=?$
(A) 1 (B) 5⁴ (C) 5⁵ (D) 5⁶
2.$(-4^{4})^{2}\div 2^{10}\div 8^{2}$
(A) 1 (B) 2 (C) 4 (D) 8

习题解答及解析

1.(D) 2.(A)
1.
${\begin{aligned}&(10\times {5^{2}})\div (2\times 5^{-3})\\=&10\times {5^{2}}\div 2\div 5^{-3}\\=&(10\div 2)\times ({5^{2}}\div 5^{-3})\\=&5\times 5^{2-(-3)}\\=&5^{1}\times 5^{5}\\=&5^{6}\end{aligned}}$
2.
${\begin{aligned}&(-4^{4})^{2}\div 2^{10}\div 8^{2}\\=&2^{16}\div 2^{10}\div 2^{6}\\=&2^{(16-10-6)}\\=&2^{0}\\=&1\end{aligned}}$