問題1: 平方根の加減
- \(3\sqrt{2}+4\sqrt{2}=7\sqrt{2}\)
- \(5\sqrt{3}-2\sqrt{3}=3\sqrt{3}\)
- \(2\sqrt{8}+3\sqrt{2}=2\sqrt{4\times2}+3\sqrt{2}=2\times2\sqrt{w}+3\sqrt{2}=7\sqrt{2}\)
- \(\sqrt{50}-\sqrt{18}=\sqrt{25\times2}-\sqrt{9\times2}=5\sqrt{2}-3\sqrt{2}=2\sqrt{2}\)
問題2: 実数の範囲
- \(x<4かつx>-4、つまり-4,x<4\)
- \(x(x-4)>0を因数分解して、解はx<0かつx>4\)
- \(x^2+3x+2=(x+1)(x+2)\leq0から、解は-2 \leq x \leq- 1\)
問題3: 平方根の乗除
- \(2\sqrt{18}\times\sqrt{2}=2\sqrt{9\times2}\times\sqrt{2}=2\times3\sqrt{2}\times\sqrt{2}=6\times2=12 \)
- \( \frac{ \sqrt{48} }{ \sqrt{3} } = \frac{ \sqrt{16 \times 3} }{ \sqrt{3} } = \frac{ 4 \sqrt{3} }{ \sqrt{3} } = 4 \)
- \( \frac{ 3 \times \sqrt{27} }{ \sqrt{3} } = \frac{ 3 \times \sqrt{9 \times 3} }{ \sqrt{3} } = 3 \times \frac{ 3 \times \sqrt{3} }{ \sqrt{3} } = 9 \)
問題4: 平方根を含む方程式
- \( \sqrt{x} + 1 = 3 \)
- \( \sqrt{x} = 2 \)
- \( x = 4 \)
- \( 2 \sqrt{x} – 5 = 3 \)
- \( 2\sqrt{x} = 8 \)
- \( \sqrt{x} = 4 \)
- \( x = 16 \)
- \( \sqrt{4x = 1} = 5 \)
- \( 4x = 1 = 25 \)
- \( 4x = 24 \)
- \( x = 6 \)
問題5: 実数の四則演算（分数と根号を含む）
- \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \)
- \( \frac{3}{4} – \frac{2}{3} = \frac{9}{12} – \frac{8}{12} = \frac{1}{12} \)
- \( \frac{ \sqrt{9} } {2} \times 3 = \frac{3}{2} \times 3 = \frac{9}{2} = 4.5 \)
- \( \frac{4}{ \sqrt{16} } = \frac{4}{4} = 1 \)
