【数学I】例題1.2.2：平方根の計算（One More）★

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% 例題I1.2.2：平方根の計算（One More）★
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次の式を計算せよ．

(1) $4\sqrt{12}-\sqrt{18}-2\sqrt{75}$

(2) $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})$

(3) $(\sqrt{5}+\sqrt{3})^2+(\sqrt{5}-\sqrt{3})^2$

(4) $(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{3}-\sqrt{5})$

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% 例題I1.2.2の解答（One More）★
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(1)

\begin{align*}
4\sqrt{12}-\sqrt{18}-2\sqrt{75}&=4\sqrt{2^2\cdot 3}-\sqrt{3^2\cdot 2}-2\sqrt{5^2\cdot 3}\\
&=8\sqrt{3}-3\sqrt{2}-10\sqrt{3}=-3\sqrt{2}-2\sqrt{3}
\end{align*}

(2) $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=(\sqrt{3})^2-(\sqrt{2})^2=3-2=1$

(3)

\begin{align*}
(\sqrt{5}+\sqrt{3})^2+(\sqrt{5}-\sqrt{3})^2&=(\sqrt{5})^2+2\sqrt{5}\sqrt{3}+(\sqrt{3})^2\\
&+(\sqrt{5})^2-2\sqrt{5}\sqrt{3}+(\sqrt{3})^2\\
&=5+2\sqrt{15}+3+5-2\sqrt{15}+3=16
\end{align*}

(4)

\begin{align*}
(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{3}-\sqrt{5})&=(\sqrt{2}+\sqrt{3})^2-(\sqrt{5})^2\\
&=2+2\sqrt{6}+3-5=2\sqrt{6}
\end{align*}

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% 問題I1.2.2：平方根の計算（One More）★
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次の式を計算せよ．

(1) $3\sqrt{20}-2\sqrt{45}+\sqrt{27}$

(2) $(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})$

(3) $(\sqrt{7}+\sqrt{2})^2-(\sqrt{7}-\sqrt{2})^2$

(4) $(\sqrt{5}+\sqrt{7}+\sqrt{3})(\sqrt{5}+\sqrt{7}-\sqrt{3})$

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% 問題I1.2.2の解答（One More）★
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(1)

\begin{align*}
3\sqrt{20}-2\sqrt{45}+\sqrt{27}&=3\sqrt{2^2\cdot 5}-2\sqrt{3^2\cdot 5}+\sqrt{3^3}\\
&=6\sqrt{5}-6\sqrt{5}+3\sqrt{3}=3\sqrt{3}
\end{align*}

(2) $(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})=(\sqrt{6})^2-(\sqrt{2})^2=6-2=4$

(3)

\begin{align*}
(\sqrt{7}+\sqrt{2})^2-(\sqrt{7}-\sqrt{2})^2&=(\sqrt{7})^2+2\sqrt{7}\sqrt{2}+(\sqrt{2})^2\\
&-\{(\sqrt{7})^2-2\sqrt{7}\sqrt{2}+(\sqrt{2})^2\}\\
&=7+2\sqrt{14}+2-(7-2\sqrt{14}+2)=4\sqrt{14}
\end{align*}

(4)

\begin{align*}
(\sqrt{5}+\sqrt{7}+\sqrt{3})(\sqrt{5}+\sqrt{7}-\sqrt{3})&=(\sqrt{5}+\sqrt{7})^2-(\sqrt{3})^2\\
&=5+2\sqrt{35}+7-3=9+2\sqrt{35}
\end{align*}

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% 例題I1.2.2の側注：平方根の計算（One More）★
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% 側注1
素因数分解し，根号内を小さい数にする．

% 側注2
$(a+b)(a-b)=a^2-b^2$

% 側注3
$(a\pm b)^2=a^2\pm 2ab+b^2$,

\[
\sqrt{a}\sqrt{b}=\sqrt{ab}(a>0,b>0)
\]

% 側注4
$(\sqrt{2}+\sqrt{3})=A$とおくと，

\[
(A+\sqrt{5})(A-\sqrt{5})=A^2-5
\]

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% 問題I1.2.2の側注：平方根の計算（One More）★
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% 側注1
素因数分解し，根号内を小さい数にする．

% 側注2
$(a+b)(a-b)=a^2-b^2$

% 側注3
$(a\pm b)^2=a^2\pm 2ab+b^2$,

\[
\sqrt{a}\sqrt{b}=\sqrt{ab}(a>0,b>0)
\]

% 側注4
$(\sqrt{5}+\sqrt{7})=A$とおくと，

\[
(A+\sqrt{3})(A-\sqrt{3})=A^2-3
\]

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% 例題I1.2.2の考え方・補足など：平方根の計算（One More）★
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% 考え方
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(1) 根号内の数を素因数分解し，$\sqrt{k^2a}=k\sqrt{a}$を用いて根号内をできるだけ小さい数にする．例えば，$\sqrt{12}$は，$\sqrt{12}=\sqrt{2^2\cdot 3}=2\sqrt{3}$とする．

(2)，(3)文字式と同じように展開して，$(\sqrt{a})^2=a,(-\sqrt{a})^2=a$を用いて計算する．

(4) 乗法公式$(a+b)(a-b)=a^2-b^2$を用いる．

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% One Point
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\[
a>0,k>0\text{のとき，}\sqrt{k^2a}=k\sqrt{a}
\]

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% 余談
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今後の学習のために，$\sqrt{a^2}=|a|=\{\begin{alignedat}{2}a&(a\geqq 0)\\-a&(a<0)\end{alignedat}$も合わせて覚えておくとよい．