在舊課綱中,該單元合併在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2-3 分數的加減及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。
a , b , m , n , x {\displaystyle a,b,m,n,x} 是不為 0 {\displaystyle 0} 的實數( a , b , m , n , x ∈ R ≠ 0 {\displaystyle a,b,m,n,x\in \mathbb {R} \neq 0} ),且a,b不可同時為負數
a m × a n = ( a × a × a × ⋯ × a ⏟ m 個 a ) × ( a × a × ⋯ × a ⏟ n 個 a ) = a × a × a × ⋯ × a ⏟ m 個 a × a × a × ⋯ × a ⏟ n 個 a = a × a × a × ⋯ × a ⏟ m + n 個 a = a m + n {\displaystyle {\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}}}
(1) if m > n : a m ÷ a n = a × a × a × ⋯ × a ⏞ m 個 a a × a × ⋯ × a ⏟ n 個 a = ⧸ a × ⧸ a × ⧸ a × ⧸ ⋯ × a × a × ⋯ × a ⏞ m 個 a ⧸ a × ⧸ a × ⧸ ⋯ × ⧸ a ⏟ n 個 a = a × a × a ⋯ a ⏟ m − n 個 a = a m − n (2) if m < n : a m ÷ a n = a × a × ⋯ × a ⏞ m 個 a a × a × a × ⋯ × a ⏟ n 個 a = ⧸ a × ⧸ a × ⧸ ⋯ × ⧸ a ⏞ m 個 a ⧸ a × ⧸ a × ⧸ a × ⧸ ⋯ × a × a × ⋯ × a ⏟ n 個 a = 1 a × a × a ⋯ a ⏟ n − m 個 a = 1 a n − m {\displaystyle {\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}}}
if m = n : a m ÷ a n = a m − n = a 0 = a × a × a × ⋯ × a ⏞ m 個 a a × a × a × ⋯ × a ⏟ n 個 a = ⧸ a × ⧸ a × ⧸ ⋯ × ⧸ a ⏞ m 個 a ⧸ a × ⧸ a × ⧸ ⋯ × ⧸ a ⏟ n 個 a = 1 ⇒ a 0 = 1 {\displaystyle {\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 × a m × a m × ⋯ × a m ⏟ n 個 a m ) = ( a × ⋯ × a ) ⏞ m × ( a × ⋯ × a ) ⏞ m × ⋯ × ( a × ⋯ × a ) ⏞ m ⏟ m × n 個 a = a × a × a × ⋯ × a ⏟ m × n 個 a = a m × n {\displaystyle {\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 × a × ⋯ × a ⏟ m 個 a ) × ( b × b × ⋯ × b ⏟ m 個 b ) = a × a × ⋯ × a ⏟ m 個 a × b × b × ⋯ × b ⏟ m 個 b = a × b × a × b × a × b × a × b × ⋯ × a × b ⏟ m 個 a 和 b 相 乘 = ( a × b ) × ( a × b ) × ( a × b ) × ( a × b ) × ⋯ × ( a × b ) ⏟ m 個 a × b = ( a × b ) m {\displaystyle {\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 × a × ⋯ × a ⏟ m 個 a ) ÷ ( b × b × ⋯ × b ⏟ m 個 b ) = a × a × ⋯ × a ⏟ m 個 a ÷ b ÷ b ÷ ⋯ ÷ b ⏟ m 個 b = a ÷ b × a ÷ b × a ÷ b × a ÷ b × ⋯ × a ÷ b ⏟ m 個 a 和 b 相 除 = ( a ÷ b ) × ( a ÷ b ) × ( a ÷ b ) × ( a ÷ b ) × ⋯ × ( a ÷ b ) ⏟ m 個 a ÷ b = ( a ÷ b ) m {\displaystyle {\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}}}
( a b ) m = ( a ÷ b ) m = a m ÷ b m = a m b m {\displaystyle {\begin{aligned}&({\frac {a}{b}})^{m}\\=&(a\div b)^{m}\\=&{a^{m}}\div {b^{m}}\\=&{\frac {a^{m}}{b^{m}}}\end{aligned}}}
a − x = ( a − 1 ) x = ( a m − n ) x ( m − n = − 1 ) = ( a m − a n ) x = ( a × a × ⋯ × a ⏞ m 個 a a × a × a × ⋯ × a ⏟ n 個 a ) x = ( a × a × ⋯ × a ⏞ m 個 a ) x ( a × a × a × ⋯ × a ⏟ n 個 a ) x = a x × a x × ⋯ × a x ⏞ m 個 a x a x × a x × a x × ⋯ × a x ⏟ n 個 a x = ⧸ a x × ⧸ a x × ⧸ ⋯ × ⧸ a x ⏞ m 個 a ⧸ a x × ⧸ a x × ⧸ a x × ⧸ ⋯ × a x ⏟ n 個 a x = 1 a x ⏟ n − m = 1 個 a x = 1 a x {\displaystyle {\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}}}
1. ( 10 × 5 2 ) ÷ ( 2 × 5 − 3 ) = ? {\displaystyle (10\times {5^{2}})\div (2\times 5^{-3})=?} (A) 1 (B) 54 (C) 55 (D) 56 2. ( − 4 4 ) 2 ÷ 2 10 ÷ 8 2 {\displaystyle (-4^{4})^{2}\div 2^{10}\div 8^{2}} (A) 1 (B) 2 (C) 4 (D) 8
1.(D) 2.(A) 1. ( 10 × 5 2 ) ÷ ( 2 × 5 − 3 ) = 10 × 5 2 ÷ 2 ÷ 5 − 3 = ( 10 ÷ 2 ) × ( 5 2 ÷ 5 − 3 ) = 5 × 5 2 − ( − 3 ) = 5 1 × 5 5 = 5 6 {\displaystyle {\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. ( − 4 4 ) 2 ÷ 2 10 ÷ 8 2 = 2 16 ÷ 2 10 ÷ 2 6 = 2 ( 16 − 10 − 6 ) = 2 0 = 1 {\displaystyle {\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}}}