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平方根の四則演算の解法 | 夢を叶える塾
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平方根の四則演算の問題の解法

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平方根が入った数の四則演算の答えを求める問題です。

基本問題

(1)\sqrt{24}+\sqrt{2}\times\sqrt{27}
\displaystyle (2)\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}
\displaystyle (3)\frac{1}{\sqrt{2}-1}+\frac{\sqrt{2}}{2-\sqrt{2}}

解き方

計算問題なのでルールに従って計算します。

解説

平方根の計算の基本は、以下の4点を抑えておきましょう。

  1. 素因数分解してルートの外に出す\sqrt{a^2}=aという性質を使う
  2. 積を分解できる\sqrt{ab}=\sqrt{a}\sqrt{b}という性質を使う
  3. ルートの中が同じ場合は文字のように扱うことができるa\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b}という性質を使う
  4. 有理化する

(1)\sqrt{24}+\sqrt{2}\times\sqrt{27}
まず1.と2.の性質で素因数分解してルートの外に出せるものは出しましょう。
\sqrt{2^3 \times 3}+\sqrt{2}\times\sqrt{3^3}
2\sqrt{2 \times 3}+3\sqrt{2}\times\sqrt{3}
2乗を外に出すことなので6は素因数分解せず残しておいても良いですね。
次に3.の性質を使います。
(2+3)\sqrt{6}=5\sqrt{6}
答えです。

\displaystyle (2)\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}
分母にルートがあり、計算の邪魔をしています。
邪魔なものには消えていただきましょう。
邪魔なものに消えて頂くのは数学の解法の王道です。

どうするかというと、\sqrt{3}+\sqrt{2}に掛けて整数になる\sqrt{3}-\sqrt{2}を分母分子に掛けます
(a-b)(a+b)=a^2-b^2の性質を使うわけですね。
このようにして分母のルートを無くすことを有理化と言いました。

1.の性質も使いながら4.の有理化をします。
\displaystyle \frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}
\displaystyle \frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}
\displaystyle \frac{\sqrt{6}(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}
\displaystyle \frac{\sqrt{2\times 3^2}-\sqrt{2^2 \times 3}}{3-2}
\displaystyle 3\sqrt{2}-2\sqrt{3}
これ以上計算できませんので、これが答えです。

\displaystyle (3)\frac{1}{\sqrt{2}-1}+\frac{\sqrt{2}}{2-\sqrt{2}}
正しい計算ルールであればどのように計算しても良いです。
有理化してから計算しても良いですし、計算した後で有理化しても良いです。

今回は計算した後で有理化してみましょう。
という事でまず通分する形で計算を進めていきます。
\displaystyle \frac{1}{\sqrt{2}-1}+\frac{\sqrt{2}}{2-\sqrt{2}}
\displaystyle \frac{1}{\sqrt{2}-1}\frac{2-\sqrt{2}}{2-\sqrt{2}}+\frac{\sqrt{2}}{2-\sqrt{2}}\frac{\sqrt{2}-1}{\sqrt{2}-1}
\displaystyle \frac{(2-\sqrt{2})+\sqrt{2}(\sqrt{2}-1)}{(\sqrt{2}-1)(2-\sqrt{2})}
\displaystyle \frac{2-\sqrt{2}+2-\sqrt{2}}{2\sqrt{2}-2-2+\sqrt{2}}
\displaystyle \frac{4-2\sqrt{2}}{3\sqrt{2}-4}
次に有理化をします。
\displaystyle \frac{4-2\sqrt{2}}{3\sqrt{2}-4}
\displaystyle \frac{(4-2\sqrt{2})(3\sqrt{2}+4)}{(3\sqrt{2}-4)(3\sqrt{2}+4)}
\displaystyle \frac{12\sqrt{2}+16-12-8\sqrt{2}}{18-16}
\displaystyle \frac{4\sqrt{2}+4}{2}
分数の約分も忘れないでください。
2\sqrt{2}+2
これが答えになります。

終わりに

平方根の四則演算は大体高校入試問題の最初の計算問題で出題されます。
二次方程式の解の公式や、三平方の定理でもルートが出てきますので、それらの計算で使う事になります。
計算のルールに沿って計算を進めるわけですが「平方数をルートの外に出すこと」と「分数になった時の約分」が忘れられがちです。
減点の対象ですから答が出ても安心せず、答を簡単な形に直せないか見直しをしましょう。

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