指数法則

指数法則

指数法則

  1. \large a^m \times a^n = a^{m+n}
  2. \large (a^m)^n = a^{m \times n}
  3. \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

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