指数法則
- \large a^m \times a^n = a^{m+n}
- \large (a^m)^n = a^{m \times n}
- \large (ab)^n = (a)^n(b)^n
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{かけ算は } & \colMM{red}{➡ 足し算に}\\ a^{m}\ \colBX{mistyrose}{$\times$}\ a^{n} &= a^{m \colBX{mistyrose}{$+$} n} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
ちなみに,こんなこともできます。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{かけ算は } & \colMM{red}{➡ 足し算に}\\ a^{m}\ \colBX{mistyrose}{$\times$}\ a^{n}\ \colBX{mistyrose}{$\times$}\ a^{k} &= a^{m\,\colBX{mistyrose}{$+$}\,n\,\colBX{mistyrose}{$+$}\,k} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
4個でも,5個でも,何個でも同じです。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{肩の数は } & \colMM{red}{➡ かける}\\ (a^{\colBX{mistyrose}{$\scriptsize m$}})^{\colBX{mistyrose}{$\scriptsize n$}} &= a^{m \colBX{mistyrose}{$\times$} n} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
ちなみに,こんなこともできます。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{肩の数は } & \colMM{red}{➡ かける}\\ \left\{\left(a^{\colBX{mistyrose}{$\scriptsize m$}}\right)^{\colBX{mistyrose}{$\scriptsize n$}}\right\}^{\colBX{mistyrose}{$\scriptsize k$}} &= a^{m\,\colBX{mistyrose}{$\times$}\,n\,\colBX{mistyrose}{$\times$}\,k} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
4個でも,5個でも,何個でも同じです。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{(中身)は } & \colMM{red}{➡ 1つずつn乗}\\ (ab)^{\colBX{mistyrose}{$\scriptsize n$}} &= (a)^{\colBX{mistyrose}{$\scriptsize n$}}\,(b)^{\colBX{mistyrose}{$\scriptsize n$}} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
ちなみに,こんなこともできます。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{(中身)は } & \colMM{red}{➡ 1つずつn乗}\\ (abc)^{\colBX{mistyrose}{$\scriptsize n$}} &= (a)^{\colBX{mistyrose}{$\scriptsize n$}}\,(b)^{\colBX{mistyrose}{$\scriptsize n$}}\,(c)^{\colBX{mistyrose}{$\scriptsize n$}} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
4個でも,5個でも,何個でも同じです。
「かけ算は足し算になる」なら、「割り算は」・・・と考えた皆さん、素晴らしいです。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{わり算は } & \colMM{red}{➡ ひき算に}\\ a^{m}\ \colBX{mistyrose}{$\div$}\ a^{n} &= a^{m \colBX{mistyrose}{$-$} n} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
ちなみに,こんなこともできます。
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{red}{わり算は } & \colMM{red}{➡ ひき算に}\\ a^{m}\ \colBX{mistyrose}{$\div$}\ a^{n}\ \colBX{mistyrose}{$\div$}\ a^{k} &= a^{m\,\colBX{mistyrose}{$-$} n\,\colBX{mistyrose}{$-$}\,k} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
次の計算をしよう
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{red}{➡整理 \Darr かけ算は}\\ 2a^3 \times 3a^4 &= 2 \times 3\ \colBX{mistyrose}{$a^3 \times a^4$}\\ & \colMM{red}{ \Darr 足し算に!}\\ &= 6 \colBX{mistyrose}{$a^{3+4}$}\\ \\ &= 6a^7 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{orange}{(中身)を } & \colMM{orange}{1つずつ \Darr 3乗 \Darr}\\ \colBX{bisque}{$(-a^2)^3$} \times a &= \colBX{bisque}{$(-1)^3$}\,\colFR{green}{\colBX{bisque}{$(a^2)^3$}} \times a\\ & \colMM{green}{ 肩の数は \Darr かける!}\\ &= -1 \colFR{green}{$a^{2 \times 3}$} \times a\\ \\ &= -1 \times \colBX{violet}{$a^6 \times a^{1}$}\\ & \colMM{magenta}{ かけ算は \Darr 足し算に}\\ &= -1 \times \colBX{violet}{$a^{6+1}$}\\ \\ &= -a^7 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{orange}{(中身)を } & \colMM{orange}{1つずつ \Darr 3乗 \Darr \Darr}\\ \colBX{bisque}{$(-3x^2y)^3$} &= \colBX{bisque}{$(-3)^3$}\,\colFR{green}{\colBX{bisque}{$(x^2)^3$}}\,\colBX{bisque}{$(y)^3$}\\ & \colMM{green}{ 肩の数は \Darr かける!}\\ &= -27 \colFR{green}{$x^{2 \times 3}$}\ y^3\\ \\ &= -27x^6y^3 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{red}{➡整理 \Darr かけ算は}\\ a^3 \times 4a^5 &= 4\ \colBX{mistyrose}{$a^3 \times a^5$}\\ & \colMM{red}{ \Darr 足し算に!}\\ &= 4 \colBX{mistyrose}{$a^{3+5}$}\\ \\ &= 4a^8 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{orange}{肩の数は } & \colMM{orange}{ かける!}\\ \colBX{bisque}{$(a^4)^2$} \times \colBX{palegreen}{$(-3a)^2$} &= \colBX{bisque}{$a^{4 \times 2}$} \times \colBX{palegreen}{$(-3)^2$}\,\colBX{palegreen}{$(a)^2$}\\ \colMM{green}{(中身)を} &\colMM{green}{ 1つずつ \nwarrow\nearrow2乗}\\ &= a^8 \times 9a^2\\ \\ &= 9\ \colBX{violet}{$a^8 \times a^2$}\\ & \colMM{magenta}{ かけ算は \Darr 足し算に}\\ &= 9\ \colBX{violet}{$a^{8+2}$}\\ \\ &= 9a^{10} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{orange}{(中身)を } & \colMM{orange}{1つずつ \Darr 3乗 \Darr \Darr}\\ \colBX{bisque}{$(-2x^3y^2)^3$} &= \colBX{bisque}{$(-2)^3$}\,\colFR{green}{\colBX{bisque}{$(x^3)^3$}}\,\colFR{green}{\colBX{bisque}{$(y^2)^3$}}\\ & \colMM{green}{ \swarrow肩の数は \Darr かける!}\\ &= -8\,\colFR{green}{$x^{3 \times 3}$}\ \colFR{green}{$y^{2 \times3}$}\\ \\ &= -8x^9y^6 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{orange}{(中身)を } & \colMM{orange}{ 1つずつ \Darr 2乗 \Darr \Darr}\\ x^2y \times \colBX{bisque}{$(-3x^3y^2)^2$} &= x^2y \times\colBX{bisque}{$(-3)^2$}\,\colFR{green}{\colBX{bisque}{$(x^3)^2$}}\,\colFR{green}{\colBX{bisque}{$(y^2)^2$}}\\ & \colMM{green}{ \swarrow肩の数は \Darr かける!}\\ &= x^2y \times 9\,\colFR{green}{$x^{3 \times 2}$}\ \colFR{green}{$y^{2 \times 2}$}\\ \\ &= x^2y \times 9x^6y^4\\ \\ &= 9\ \colBX{violet}{$x^2 \times x^6$}\ \colBX{violet}{$y^1 \times y^4$}\\ & \colMM{magenta}{ \Darr かけ算は \Darr 足し算に!}\\ &= 9\,\colBX{violet}{$x^{2+6}$}\,\colBX{violet}{$y^{1+4}$}\\ \\ &= 9x^8y^5 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} & \colMM{red}{➡整理 \Darr かけ算は}\\ -x^2 \times 3x^3 &= -3\ \colBX{mistyrose}{$x^2 \times x^3$}\\ & \colMM{red}{ \Darr 足し算に!}\\ &= -3 \colBX{mistyrose}{$x^{2+3}$}\\ \\ &= -3x^5 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{green}{肩の数は} & \colMM{green}{ かける!}\\ \colBX{bisque}{$(-2a)^3$} \times \colBX{palegreen}{$(a^4)^3$} &= \colBX{bisque}{$(-2)^3$}\,\colBX{bisque}{$(a)^3$} \times \colBX{palegreen}{$a^{4 \times 3}$}\\ \colMM{orange}{(中身)を } &\colMM{orange}{ 1つずつ \nwarrow\nearrow 3乗}\\ &= -8a^3 \times a^{12}\\ \\ &= -8\ \colBX{violet}{$a^3 \times a^{12}$}\\ & \colMM{magenta}{ かけ算は \Darr 足し算に}\\ &= -8\ \colBX{violet}{$a^{3+12}$}\\ \\ &= -8a^{15} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize\bf\bm #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \begin{align*} \colMM{orange}{(中身)を } & \colMM{orange}{1つずつ \Darr 2乗 \Darr \Darr}\\ \colBX{bisque}{$(5x^2y)^2$} &= \colBX{bisque}{$(5)^2$}\,\colFR{green}{\colBX{bisque}{$(x^2)^2$}}\,\colFR{green}{\colBX{bisque}{$(y^1)^2$}}\\ & \colMM{green}{ \swarrow肩の数は \Darr かける!}\\ &= 25\,\colFR{green}{$x^{2 \times 2}$}\ \colFR{green}{$y^{1 \times 2}$}\\ \\ &= 25x^4y^2 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
レベルG
次の計算をせよ。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & 54a^2b^7 \times (-2a^3b)^2 \div (-3ab^2)^3\\ & \colMM{red}{ 1つずつ2乗\Darr \Darr \Darr}\colMM{red}{ \Darr \Darr \Darr 1つずつ3乗}\\ =\ & 54a^2b^7 \times (-2)^2(a^{\colBX{bisque}{$\scriptsize 3$}})^{\colBX{bisque}{$\scriptsize 2$}}(b)^2 \div (-3)^3(a)^3(b^{\colBX{palegreen}{$\scriptsize 2$}})^{\colBX{palegreen}{$\scriptsize 3$}}\\ & \colMM{orange}{\bf \swarrowかける}\colMM{green}{\bf \swarrowかける}\\ =\ & 54a^2b^7 \times 4a^{\colBX{bisque}{$\scriptsize 6$}}b^2 \div (-27a^3b^{\colBX{palegreen}{$\scriptsize 6$}})\\ & \colMM{magenta}{\bf \div 割り算は}\\ =\ & \dfrac{54a^2b^7 \times 4a^6b^2}{-27a^3b^6}\colMM{magenta}{\bf \swarrow 分母へ}\\ \\ =\ & \dfrac{2 \cdot 27 \times 4 a^8b^9}{-1 \cdot 27a^3b^6}\\ \\ =\ & \dfrac{-8 a^8b^9}{a^3b^6} = -8a^5b^3 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} & (-2x^2y)^3 \times 3xy^2 \div 6x^3y^2\\ & \colMM{red}{1つずつ2乗\Darr}\\ =\ & (-2)^3(x^{\colBX{bisque}{$\scriptsize 2$}})^{\colBX{bisque}{$\scriptsize 3$}}(y)^3 \times \times 3xy^2 \div 6x^3y^2\\ & \colMM{orange}{\bf \swarrowかける}\\ =\ & -8x^{\colBX{bisque}6}y^3 \times 3xy^2 \div 6x^3y^2\\ & \colMM{magenta}{\bf \div 割り算は}\\ =\ & \dfrac{-8x^{\colBX{bisque}6}y^3 \times 3xy^2}{6x^3y^2}\colMM{magenta}{\bf \swarrow 分母へ}\\ \\ =\ & \dfrac{-4 \cdot 2 \times3 x^7y^5}{6x^3y^2}\\ \\ =\ & \dfrac{-4 x^7y^5}{x^3y^2} = -4x^4y^3 \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan