[Title]
102B_数の計算（分数，平方根）

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[Problem]
次の式の計算をせよ。




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5


[Level1]
分数の計算（超基礎）
$-\displaystyle \frac{8}{9} \div \displaystyle \frac{3}{4}$

$-\displaystyle \frac{32}{27}$

$-\displaystyle \frac{1}{5}+\displaystyle \frac{3}{4}$

$\displaystyle \frac{11}{20}$

$\left(-\displaystyle \frac{5}{6} \right) \div \left(-\displaystyle \frac{3}{10} \right)$

$\displaystyle \frac{25}{9}$

$-\displaystyle \frac{2}{5}+\displaystyle \frac{4}{3}$

$\displaystyle \frac{14}{15}$

$8 \div \displaystyle \frac{4}{5}$

$10$

$\left( -\displaystyle \frac{5}{8} \right) \times \left( -\displaystyle \frac{4}{3} \right)$

$\displaystyle \frac{5}{6}$

$\left( -\displaystyle \frac{3}{7} \right) \times \displaystyle \frac{14}{6}$

$-1$

$\displaystyle \frac{13}{2} \div \left( -\displaystyle \frac{5}{6} \right)$

$-\displaystyle \frac{39}{5}=-7 \displaystyle \frac{4}{5}$

$\left( -\displaystyle \frac{11}{3} \right) \div \left( -\displaystyle \frac{33}{6} \right)$

$\displaystyle \frac{2}{3}$

$\displaystyle\frac{3}{7}-\left( -\displaystyle\frac{4}{7} \right)$

1








[Level2]
分数の計算（基礎）
$4 \div \displaystyle \frac{2}{3} \times \left( -\displaystyle \frac{5}{6} \right)$

$-5$

$\displaystyle \frac{1}{4} \times 20 \div \displaystyle \frac{5}{16}$

$16$

$-\displaystyle \frac{3}{4}+\displaystyle \frac{1}{4}-1$

$-\displaystyle \frac{3}{2}$

$\left(-\displaystyle \frac{3}{5} \right)-\left(-\displaystyle \frac{4}{3} \right)$

$\displaystyle \frac{11}{15}$

$\left(-\displaystyle \frac{4}{7} \right)-\left(+\displaystyle \frac{2}{3} \right)$

$-\displaystyle \frac{26}{21}$

$\left( -\displaystyle \frac{2}{3} \right)+\left( -\displaystyle \frac{1}{2} \right)-\left( +\displaystyle \frac{5}{6} \right)$

$-2$

$\displaystyle \frac{3}{4}-\left( -\displaystyle \frac{1}{2} \right)-\displaystyle \frac{1}{4}$

$1$

$\displaystyle \frac{1}{3}-\left( \displaystyle \frac{1}{4}-\displaystyle \frac{5}{3} \right)$

$\displaystyle \frac{7}{4}$

$(-1.8)\div 0.4$

process
$-\displaystyle \frac{18}{10} \div \displaystyle \frac{4}{10}=-\displaystyle \frac{18}{10} \times \displaystyle \frac{10}{4}$

$-\displaystyle \frac{9}{2} $

$0.75 \div (-2.5)$

process
$\displaystyle \frac{3}{4} \div \left( -\displaystyle \frac{5}{2} \right)=\displaystyle \frac{3}{4} \times \left( -\displaystyle \frac{2}{5} \right)$

$-\displaystyle \frac{3}{10} $

$\displaystyle \frac{1}{2}-\displaystyle \frac{3}{4} \div \displaystyle \frac{9}{10}$

process
$\displaystyle \frac{1}{2}-\displaystyle \frac{3}{4} \times \displaystyle \frac{10}{9}$

$-\displaystyle \frac{1}{3}$

$\displaystyle \frac{5}{8}- \left( -\frac{3}{2} \right)^{2}$

process
$\displaystyle \frac{5}{8}-\displaystyle \frac{9}{4}$

$-\displaystyle \frac{13}{8}$

$\displaystyle 2-\left( -\frac{1}{2} \right)^2 \times 24$

process
$2-\displaystyle \frac{1}{4} \times 24$

$-4$

$6 \times \left( -\displaystyle \frac{3}{4} \right)-\displaystyle \frac{5}{4}$

$-\displaystyle \frac{23}{4}$

$\left( \displaystyle \frac{1}{3} - \displaystyle \frac{2}{5} \right) \times \displaystyle \frac{5}{2}$

$-\displaystyle \frac{1}{6}$

$2 \times \left( \displaystyle \frac{1}{4} - \displaystyle \frac{1}{3} \right)$

$-\displaystyle \frac{1}{6}$

$\left( \displaystyle \frac{5}{3} - \displaystyle \frac{3}{4} \right) \div \displaystyle \frac{11}{3}$

$\displaystyle \frac{1}{4}$

$\displaystyle \frac{2}{3} + \displaystyle \frac{1}{2} \times \left( -\displaystyle \frac{2}{5} \right)$

$\displaystyle \frac{7}{15}$

$\displaystyle \frac{1}{3} \div \displaystyle \frac{5}{8} \times \left(-\displaystyle \frac{1}{3} \right)$

$-\displaystyle \frac{8}{45}$

$\displaystyle \frac{3}{2} \times \displaystyle \frac{7}{8} \div \displaystyle \frac{3}{4} \times \displaystyle \frac{1}{2}$

process
$\displaystyle \frac{3}{2} \times \displaystyle \frac{7}{8} \times \displaystyle \frac{4}{3} \times \displaystyle \frac{1}{2}$

$\displaystyle \frac{7}{8} $

$\displaystyle\frac{3}{4}-\displaystyle\frac{4}{5}-\left( -\displaystyle\frac{1}{2} \right)$

process
$\displaystyle\frac{3}{4}-\displaystyle\frac{4}{5}+\displaystyle\frac{2}{4}=\displaystyle\frac{5}{4}-\displaystyle\frac{4}{5}
=\displaystyle\frac{9}{20}$

$\displaystyle\frac{9}{20}$

$\displaystyle\frac{5}{12}\div \displaystyle\frac{10}{27}\times \displaystyle\frac{4}{9}$

process
$\displaystyle\frac{5}{12}\times \displaystyle\frac{27}{10}\times \displaystyle\frac{4}{9}$

$\displaystyle\frac{1}{2}$

$\displaystyle\frac{5}{6}\times (-3)\div \left( -\displaystyle\frac{1}{2} \right)$

process
$\displaystyle\frac{5}{6}\times (-3)\times ( -2)$

$5$

$\displaystyle\frac{9}{14}\times\displaystyle\frac{7}{8}\times\displaystyle\frac{2}{3}$

$\displaystyle\frac{3}{8}$

$\displaystyle\frac{7}{10}\div\displaystyle\frac{7}{15}\div\displaystyle\frac{5}{3}$

process
$\displaystyle\frac{7}{10}\times\displaystyle\frac{15}{7}\times\displaystyle\frac{3}{5}$

$\displaystyle\frac{9}{10}$

$\displaystyle\frac{5}{4}\div 0.4 \times\displaystyle\frac{8}{5}$

process
$\displaystyle\frac{5}{4}\times\displaystyle\frac{10}{4}\times\displaystyle\frac{8}{5}$

$5$

$\displaystyle\frac{1}{2}+\left( -\displaystyle\frac{1}{6} \right)\times 5$

process
$\displaystyle\frac{3}{6}-\displaystyle\frac{5}{6} $

$-\displaystyle\frac{1}{3}$

$\left( -\displaystyle\frac{1}{6}\right)\times 4-\left( -\displaystyle\frac{2}{3} \right)\times 4$

process
$\left( -\displaystyle\frac{1}{6}+\displaystyle\frac{2}{3}\right)\times 4 
=\displaystyle\frac{1}{2}\times 4$

$2$

$2+\displaystyle\frac{1}{2} \div \left( -\displaystyle\frac{3}{4} \right)$

process
$2+\displaystyle\frac{1}{2} \times \left( -\displaystyle\frac{4}{3} \right)=2-\displaystyle\frac{2}{3}=\displaystyle\frac{6-2}{3}$

$\displaystyle\frac{4}{3}$

$\displaystyle\frac{5}{6}-\left( -\displaystyle\frac{3}{2} \right)^2$

process
$\displaystyle\frac{5}{6}-\displaystyle\frac{9}{4}$

$-\displaystyle\frac{17}{12}$

$\displaystyle\frac{2}{9}\times \left( -\displaystyle\frac{15}{4} \right)\div \displaystyle\frac{5}{6}$

process
$-\displaystyle\frac{5}{6}\times\displaystyle\frac{6}{5}=-1$

$-1$

$4\times \displaystyle\frac{3}{2}+(-3)$

process
$2\times 3-3$

3

$-\displaystyle\frac{2}{3}+\displaystyle\frac{1}{4}
-\displaystyle\frac{1}{6}$

$-\displaystyle\frac{7}{12}$

$\displaystyle\frac{2}{3}\times \left( -\displaystyle\frac{2}{15} \right)\div \displaystyle\frac{2}{3}$

$-\displaystyle\frac{1}{6}$

$\displaystyle\frac{3}{5}\div \displaystyle\frac{4}{5} \times 
\left( -\displaystyle\frac{2}{3} \right)$

$-\displaystyle\frac{1}{2}$

$\displaystyle\frac{1}{1\times 2}+\displaystyle\frac{1}{2\times 3}+\displaystyle\frac{1}{3\times 4}$\\
\textcircled{\small 1}\,$\displaystyle\frac{1}{2}$~~~
\textcircled{\small 2}\,$\displaystyle\frac{7}{12}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{2}{3}$~~~
\textcircled{\small 4}\,$\displaystyle\frac{3}{4}$~~~
\textcircled{\small 5}\,$\displaystyle\frac{5}{6}$

process
$\displaystyle\frac{4}{6}+\displaystyle\frac{1}{3\times 4}=\displaystyle\frac{3}{4}$

\textcircled{\small 4}


















[Level3]
分数の計算（応用）
$\displaystyle\frac{2}{3}\times \displaystyle\frac{1}{5} \div \displaystyle\frac{1}{3}+\displaystyle\frac{1}{4}$

$\displaystyle\frac{13}{20}$

$3 \displaystyle \frac{3}{4} \times 0.35 \div 3 \displaystyle \frac{1}{2}$

$\displaystyle \frac{3}{8}$

$(-20)\div\displaystyle \frac{4}{3} \times \left( -\displaystyle \frac{8}{5}\right)$

$24$

$1.25-\left( -\displaystyle \frac{2}{3}+\displaystyle \frac{3}{4} \right)$

$\displaystyle \frac{7}{6}$

$\left( \displaystyle \frac{1}{2}-\displaystyle \frac{2}{3} \right)\times (-6)-\left( -\displaystyle \frac{1}{4} \right) \div \left( -\displaystyle \frac{1}{2} \right)$

$\displaystyle \frac{1}{2}$

$\left( -\displaystyle \frac{3}{5} \right) \times \left( -\displaystyle \frac{10}{9} \right)\div \displaystyle \frac{2}{3}$

$1$

$\left( -\displaystyle \frac{2}{3} \displaystyle \right )^{2} \times \displaystyle \frac{2}{5}\div \left( -\frac{4}{15} \right )$

process
$\displaystyle \frac{4}{9} \times \displaystyle \frac{2}{5} \times \left( -\displaystyle \frac{15}{4} \right )$

$-\displaystyle \frac{2}{3}$

$\displaystyle \left( -\frac{1}{4} \right ) \times \frac{3}{5} \div \left( -\frac{3}{2} \right)$

process
$\displaystyle \left( -\frac{1}{4} \right ) \times \frac{3}{5} \times \left( -\frac{2}{3} \right)$

$\displaystyle \frac{1}{10}$

$\displaystyle \frac{1}{6} \div \left( 1.5 -\frac{1}{3} \right)$

process
$\displaystyle \frac{1}{6} \div \left( \frac{3}{2}-\frac{1}{3} \right)
=\displaystyle \frac{1}{6} \div \left( \frac{9}{6}-\frac{2}{6} \right)$

$\displaystyle \frac{1}{7}$

$\left( -\displaystyle \frac{2}{3} \right) \times \displaystyle \frac{7}{8}+\displaystyle \frac{1}{6}$  

$-\displaystyle \frac{5}{12}$ 

$\displaystyle \frac{3}{2}+\displaystyle \frac{2}{3} \div \left( -\displaystyle \frac{4}{3} \right)$

$1$

$2-6 \div \left( -\displaystyle \frac{2}{3} \right)$

$11$

$\displaystyle \frac{1}{4} - \left( -\displaystyle \frac{7}{6} \right) \div \displaystyle \frac{7}{2}$

$\displaystyle \frac{7}{12}$

$\displaystyle \frac{1}{2} \times \left( -\displaystyle \frac{4}{3} \right)+ \displaystyle \frac{1}{5}$

$-\displaystyle \frac{7}{15}$

$\left( -\displaystyle \frac{3}{10} \right) \div \displaystyle \frac{5}{2} \times \displaystyle \frac{5}{3}$

$-\displaystyle \frac{1}{5}$

$\left( \displaystyle \frac{1}{2} - \displaystyle \frac{4}{3} \right) \times \displaystyle \frac{3}{10}$

$-\displaystyle \frac{1}{4}$

$0.004 \times 15 \div (0.05 \times 0.6)$

process
$\displaystyle \frac{0.004 \times 15}{0.05 \times 0.6}$\\
分母，分子に$1000$をかけると，$\displaystyle \frac{4 \times 15}{5 \times 6}=2$

$2$

$-\displaystyle\frac{3}{1+\left( -\displaystyle\frac{1}{4} \right)^2 \times 8}$

$-2$

$\left( -\displaystyle\frac{1}{2} \right)^2\times 4-7$

$-6$

$-(-2)^2-\left( -\displaystyle\frac{1}{3} \right)\times \displaystyle\frac{3}{2}$

$-\displaystyle\frac{7}{2}$

$-14-(-15)\div \left( -\displaystyle\frac{3}{2} \right)$

$-24$

$4^2\div (-2)-\displaystyle\frac{3}{2}\times (-8)$

process
$\displaystyle\frac{16}{-2}+3\times 4=-8+12$

4





[Level4]
平方根の簡単化（基礎）
次の式を簡単にしなさい。\\
$\sqrt{3} \times \sqrt{6}$

process
$\sqrt{18}=\sqrt{3^2 \times 2}$

$3\sqrt{2}$

次の式を簡単にしなさい。\\
$\sqrt{5} \times \sqrt{8}$

process
$\sqrt{40}=\sqrt{2^2 \times 10}$

$2 \sqrt{10}$

次の式を簡単にしなさい。\\
$2\sqrt{6} \times \sqrt{20}$

process
$2\sqrt{4 \times 6 \times 5}=2 \sqrt{2^2 \times 30}$

$4\sqrt{30}$

次の式を簡単にしなさい。\\
$\sqrt{24} \div \sqrt{3}$

process
$\displaystyle \frac{\sqrt{24}}{\sqrt{3}}=\sqrt{8}$

$2\sqrt{2}$

次の式を簡単にしなさい。\\
$\sqrt{125} \div \sqrt{5}$

process
$\displaystyle \frac{\sqrt{125}}{\sqrt{5}}=\sqrt{25}$

$5$

次の式を簡単にしなさい。\\
$\sqrt{90} \div \sqrt{5}$

process
$\displaystyle \frac{\sqrt{90}}{\sqrt{5}}=\sqrt{18}=\sqrt{3^2 \times 2}$

$3\sqrt{2}$

次の式を簡単にしなさい。\\
$3 \sqrt{2}+4 \sqrt{2}-5 \sqrt{2}$

$2 \sqrt{2}$

次の式を簡単にしなさい。\\
$\sqrt{5}+3 \sqrt{7}-3\sqrt{5}+2 \sqrt{7}$

$5\sqrt{7}-2\sqrt{5}$

次の式を簡単にしなさい。\\
$(\sqrt{18}+\sqrt{6}) \div \sqrt{2}$

process
$\displaystyle \frac{\sqrt{18}+\sqrt{6}}{\sqrt{2}}=\sqrt{9}+\sqrt{3}$

$3+\sqrt{3}$

次の式を簡単にしなさい。\\
$(\sqrt{27}-2 \sqrt{15}) \div \sqrt{3}$

process
$\displaystyle \frac{\sqrt{27}-2 \sqrt{15}}{\sqrt{3}}=\sqrt{9}-2 \sqrt{5}$

$3-2 \sqrt{5}$

次の式を簡単にしなさい。\\
$4\sqrt{2}-4-\sqrt{2}+5$

$3\sqrt{2}+1$

次の式を簡単にしなさい。\\
$4\sqrt{3}-\sqrt{12}$

$2\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{5}\times\sqrt{20}$

process
$\sqrt{5}\times\sqrt{4\times 5}=\sqrt{5^2\times 2^2}=10$

$10$

次の式を簡単にしなさい。\\
$\sqrt{18}\div\sqrt{6}$

process
$\sqrt{\displaystyle\frac{18}{6}}=\sqrt{3}$

$\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{27}+\displaystyle\frac{6}{\sqrt{3}}$

process
$\sqrt{9\times 3}+\displaystyle\frac{6\sqrt{3}}{3}=3\sqrt{3}+2\sqrt{3}$

$5\sqrt{3}$

次の計算をせよ。\\
$\sqrt{50}+\sqrt{8}$

process
$5\sqrt{2}+2\sqrt{2}=7\sqrt{2}$

$7\sqrt{2}$

次の式を簡単にしなさい。\\
$2\sqrt{32}-\sqrt{8}$

process
$2\times 2\sqrt{8}-\sqrt{8}=3\sqrt{8}$

$6\sqrt{2}$

次の式を簡単にしなさい。\\
$\sqrt{5}+3\sqrt{20}$

process
$\sqrt{5}+3\times 2\sqrt{5}=7\sqrt{5}$

$7\sqrt{5}$











[Level5]
平方根の簡単化（3項，+-）
次の式を簡単にしなさい。\\
$\sqrt{27}+4\sqrt{12}-3\sqrt{3}$

process
$3\sqrt{3}+4\times 2\sqrt{3}-3\sqrt{3}=8\sqrt{3}$

$8\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{20}-\sqrt{80}+3\sqrt{5}$

process
$\sqrt{20}-2\sqrt{20}+3\sqrt{5}=-\sqrt{20}+3\sqrt{5}\\
=-2\sqrt{5}+3\sqrt{5}=\sqrt{5}$

$\sqrt{5}$

$\sqrt{27}+2\sqrt{12}-\sqrt{3}$

process
$3\sqrt{3}+2\times 2\sqrt{3}-\sqrt{3}=6\sqrt{3}$

$6\sqrt{3}$

$\sqrt{3}-\sqrt{12}+\displaystyle\frac{9}{\sqrt{3}}\\
=\sqrt{3}-2\sqrt{3}+3\sqrt{3}=2\sqrt{3}$

$2\sqrt{3}$

$\sqrt{45}-\displaystyle\frac{15}{\sqrt{5}}-\sqrt{20}\\
=3\sqrt{5}-3\sqrt{5}-2\sqrt{5}=-2\sqrt{5}$

$-2\sqrt{5}$

$\sqrt{8}+\sqrt{3}+2\sqrt{18}$

process
$2\sqrt{2}+\sqrt{3}+2 \times \sqrt{3^2 \times 2}=2\sqrt{2}+\sqrt{3}+6 \sqrt{2}$

$8\sqrt{2}+\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{75}-2\sqrt{48}+6 \sqrt{12}$

process
$\sqrt{25 \times 3}-2\sqrt{16 \times 3}+6 \sqrt{4 \times 3}\\
=\sqrt{5^2 \times 3}-2 \sqrt{4^2 \times 3}+6 \sqrt{2^2 \times 3}\\
=5\sqrt{3}-8\sqrt{3}+12\sqrt{3}$

$9\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{125}-3\sqrt{5}+\sqrt{20}$

process
$\sqrt{25\times 5}-3\sqrt{5}+\sqrt{4 \times 5}\\
=\sqrt{5^2 \times 5}-3\sqrt{5}+\sqrt{2^2 \times 5}\\
=5 \sqrt{5}-3\sqrt{5}+2\sqrt{5}=4\sqrt{5}$

$4 \sqrt{5}$

次の式を簡単にしなさい。\\
$3\sqrt{24}+\sqrt{6}+\sqrt{150}$

process
$3\sqrt{4 \times 6}+\sqrt{6}+\sqrt{25 \times 6}\\
=3\sqrt{2^2 \times 6}+\sqrt{6}+\sqrt{5^2 \times 6}\\
6\sqrt{6}+\sqrt{6}+5 \sqrt{6}$

$12\sqrt{6}$

次の式を簡単にしなさい。\\
$2 \sqrt{3}(\sqrt{18}-2\sqrt{12})$

process
$2 \sqrt{3}(\sqrt{9 \times 2}-2 \sqrt{4 \times 3})\\
=2 \sqrt{3}(\sqrt{3^2 \times 2}- 2 \sqrt{2^2 \times 3})
=2 \sqrt{3}(3\sqrt{2}- 4 \sqrt{3})$

$6 \sqrt{6}-24$

次の式を簡単にしなさい。\\
$(1-\sqrt{3})^2$

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

$4-2\sqrt{3}$

次の式を簡単にしなさい。\\
$(1+\sqrt{5})^2$

$6+2 \sqrt{5}$

次の式を簡単にしなさい。\\
$(\sqrt{5}+1)(\sqrt{5}-1)$

process
$(\sqrt{5})^2-1^2=5-1$

$4$

次の式を簡単にしなさい。\\
$\sqrt{\displaystyle \frac{7}{4}}+\sqrt{\displaystyle \frac{7}{64}}-\sqrt{7}$

process
$\displaystyle \frac{\sqrt{7}}{2}+\displaystyle \frac{\sqrt{7}}{8}-\sqrt{7}
=\left( \displaystyle \frac{1}{2}+ \displaystyle \frac{1}{8}-1 \right) \sqrt{7}$

$-\displaystyle \frac{3\sqrt{7}}{8}$

次の式を簡単にしなさい。\\
$(2\sqrt{6}+1)(3 \sqrt{3}+\sqrt{2})$

process
$6 \sqrt{18}+2 \sqrt{12}+3 \sqrt{3}+\sqrt{2}\\
6 \sqrt{3^2 \times 2}+2 \sqrt{2^2 \times 3}+3 \sqrt{3}+\sqrt{2}\\
=18 \sqrt{2}+4 \sqrt{3}+3 \sqrt{3}+\sqrt{2}$

$19 \sqrt{2}+7 \sqrt{3}$

次の式を簡単にしなさい。\\
$(3+2\sqrt{3})(3-2 \sqrt{3})$

process
$3^2-(2\sqrt{3})^2=9-4 \times 3$

$-3$

次の式を簡単にしなさい。\\
$(\sqrt{3}+1)(\sqrt{3}-1)$

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

$2$

次の式を簡単にしなさい。\\
$\sqrt{18}-\sqrt{32}+5\displaystyle \frac{1}{\sqrt{2}}$

process
$3 \sqrt{2}-4 \sqrt{2}+\displaystyle \frac{5}{2} \sqrt{2}$

$\displaystyle \frac{3}{2} \sqrt{2}$

次の式を簡単にしなさい。\\
$(\sqrt{5}+2)(\sqrt{5}-2)$

process
$(\sqrt{5})^2-2^2=5-4$

$1$

次の式を簡単にしなさい。\\
$(\sqrt{5}+3)(\sqrt{5}-3)$

process
$(\sqrt{5})^2-3^2=-4$

$-4$

次の式を簡単にしなさい。\\
$3\sqrt{12}-\displaystyle \frac{24}{\sqrt{3}}+\sqrt{27}$

process
$3 \times 2 \sqrt{3}-24\times \displaystyle \frac{\sqrt{3}}{3}+3 \sqrt{3}$

$\sqrt{3}$

次の式を簡単にしなさい。\\
$(1+\sqrt{2})^2$

$3+2\sqrt{2}$

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

$2+4\sqrt{3}$

次の式を簡単にせよ。\\
$\sqrt{75}-\displaystyle\frac{27}{\sqrt{3}}$

process
$5\sqrt{3}-9\sqrt{3}$

$-4\sqrt{3}$

次の式を簡単にせよ。\\
$(\sqrt{5}-\sqrt{8})^2+\sqrt{10}$

process
$13-2\sqrt{5 \times 8}+\sqrt{10}
=13-4\sqrt{10}+\sqrt{10}$

$13-3\sqrt{10}$

次の式を簡単にしなさい。\\
$\sqrt{32}-\sqrt{18}+\sqrt{8}$

process
$\sqrt{16\times 2}-\sqrt{9\times 2}+\sqrt{4\times 2}\\
=4\sqrt{2}-3\sqrt{2}+2\sqrt{2}=3\sqrt{2}$

$3\sqrt{2}$

次の式を簡単にしなさい。\\
$\sqrt{27}+\sqrt{12}-\sqrt{48}$

process
$\sqrt{9\times 3}+\sqrt{4\times 3}-\sqrt{16\times 3}\\
=3\sqrt{3}+2\sqrt{3}-4\sqrt{3}=\sqrt{3}$

$\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{\displaystyle\frac{32}{9}}-\sqrt{\displaystyle\frac{8}{9}}$

process
$\sqrt{\displaystyle\frac{16\times 2}{9}}-\sqrt{\displaystyle\frac{4\times 2}{9}}\\
=\displaystyle\frac{4\sqrt{2}}{3}-\displaystyle\frac{2\sqrt{2}}{3}$

$\displaystyle\frac{2\sqrt{2}}{3}$

次の式を簡単にしなさい。\\
$2\sqrt{3}\times\sqrt{6}\div\sqrt{24}$

process
$\displaystyle\frac{2\sqrt{3}\times\sqrt{6}}{\sqrt{24}}
=2\sqrt{\displaystyle\frac{3 \times 6}{4 \times 6}}=\sqrt{3}$

$\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{50}-\displaystyle\frac{20}{\sqrt{2}}$

process
$\sqrt{25\times 2}-\displaystyle\frac{20\sqrt{2}}{2}=5\sqrt{2}-10\sqrt{2}$

$-5\sqrt{2}$

次の式を簡単にしなさい。\\
$(\sqrt{3}+\sqrt{5})^2$

process
$=3+5+2\sqrt{15}$

$8+2\sqrt{15}$

次の式を簡単にしなさい。\\
$(\sqrt{7}-\sqrt{3})^2$

process
$=7+3-2\sqrt{21}$

$10-2\sqrt{21}$

次の式を簡単にしなさい。\\
$(\sqrt{6}+4)(\sqrt{6}-4)$

process
$6-4^2=6-16=-10$

$-10$

次の式を簡単にしなさい。\\
$(5+2\sqrt{2})(5-2\sqrt{2})$

process
$5^2-(2\sqrt{2})^2=25-4\times 2$

$17$

次の式を簡単にしなさい。\\
$(2\sqrt{3}+3\sqrt{2})^2-12\sqrt{6}$

process
$4\times 3+9\times 2+12\sqrt{6}-12\sqrt{6}=12+18$

$30$

次の式を簡単にしなさい。\\
$(\sqrt{27}+3\sqrt{2})(2\sqrt{3}-\sqrt{8})$

process
$(\sqrt{9\times 3}+3\sqrt{2})(2\sqrt{3}-\sqrt{4\times 2})\\
=(3\sqrt{3}+3\sqrt{2})(2\sqrt{3}-2\sqrt{2})\\
=3\times (\sqrt{3}+\sqrt{2})\times 2\times (\sqrt{3}-\sqrt{2})\\
=6\times (3-2)$

6

次の式を簡単にしなさい。\\
$(\sqrt{3}+\sqrt{2})^2(\sqrt{3}-\sqrt{2})^2$

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

1

次の式を簡単にしなさい。\\
$(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{3}-\sqrt{5})$

process
$\left( \sqrt{2}+\sqrt{3} \right)^2-5=2+3+2\sqrt{6}-5$

$2\sqrt{6}$

次の式を簡単にしなさい。\\
$\sqrt{8}+\sqrt{18}-\sqrt{27}-\sqrt{12}$

process
$2\sqrt{2}+3\sqrt{2}-3\sqrt{3}-2\sqrt{3}
=5\sqrt{2}-5\sqrt{3}$

$5\sqrt{2}-5\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{32}\div 2\sqrt{2}\times (-\sqrt{2})$

process
$\displaystyle\frac{\sqrt{32}\times(-\sqrt{2})}{2\sqrt{2}}=-\displaystyle\frac{32}{4}=-\sqrt{8}=-2\sqrt{2}$

$-2\sqrt{2}$

次の式を簡単にしなさい。\\
$\sqrt{3} \left( \sqrt{3}+2 \right)-2\sqrt{3}$

$3$

次の式を簡単にしなさい。\\
$\left( \sqrt{3}+2 \right)^2-\left( \sqrt{3}-2 \right)^2$

process
$\{ \left( \sqrt{3}+2 \right)-\left( \sqrt{3}-2 \right) \}\{ \left( \sqrt{3}+2 \right)+\left( \sqrt{3}-2 \right) \}\\
=4 \times 2\sqrt{3}$

$8\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{5} \left( \sqrt{80}-3\sqrt{5} \right)$

process
$\sqrt{5} \times \sqrt{5 \times 16}-3\sqrt{5 \times 5}$
$=5 \times 4-3 \times 5=5$

$5$

次の式を簡単にしなさい。\\
$\sqrt{6} \left( \sqrt{8}-\displaystyle\frac{1}{\sqrt{2}} \right)$

process
$\sqrt{6} \times \sqrt{8}-\displaystyle\frac{\sqrt{6}}{\sqrt{2}}\\
=\sqrt{2\times 3\times 2 \times 4}-\sqrt{3}\\
=4\sqrt{3}-\sqrt{3}=3\sqrt{3}$

$3\sqrt{3}$

次の式を計算した値を，
\textcircled{\small 1}$\sim$\textcircled{\small 5}の中から選び
記号で答えなさい。\\
$\displaystyle\frac{3}{\sqrt{5}-\sqrt{2}}-\sqrt{5}$\\
\textcircled{\small 1}\,$-\sqrt{2}$~~~
\textcircled{\small 2}\,$-\sqrt{5}+\sqrt{2}$~~~
\textcircled{\small 3}\,$\sqrt{2}$~~~
\textcircled{\small 4}\,$\sqrt{5}-\sqrt{2}$~~~
\textcircled{\small 5}\,$\sqrt{5}$

process
$(\sqrt{5}+\sqrt{2})-\sqrt{5}=\sqrt{2}$

\textcircled{\small 3}















[Level6]
分母の有理化（+-以外）
次の式の分母を有理化し，簡単にしなさい。\\
$5\sqrt{12}-\displaystyle\frac{24}{\sqrt{3}}$

process
$5\times 2\sqrt{3}-\displaystyle\frac{24 \sqrt{3}}{3}=10\sqrt{3}-8\sqrt{3}=2\sqrt{3}$

$2\sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{12}+\displaystyle \frac{1}{\sqrt{3}}$

process
$2 \sqrt{3}+\displaystyle \frac{\sqrt{3}}{3}$

$\displaystyle \frac{7}{3} \sqrt{3}$

次の式を簡単にしなさい。\\
$\sqrt{8}+\displaystyle \frac{1}{\sqrt{2}}$

process
$2\sqrt{2}+\displaystyle \frac{\sqrt{2}}{2}=\displaystyle \frac{5\sqrt{2}}{2}$

$\displaystyle \frac{5\sqrt{2}}{2}$

次の式を簡単にしなさい。\\
$\sqrt{18}+\displaystyle \frac{8}{\sqrt{2}}$

process
$3\sqrt{2}+8 \displaystyle \frac{\sqrt{2}}{2}=3\sqrt{2}+4\sqrt{2}$

$7\sqrt{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{5}{\sqrt{2}}-\displaystyle \frac{2}{\sqrt{2}}$

process
$\displaystyle \frac{3}{\sqrt{2}}=\displaystyle \frac{3\sqrt{2}}{2}$

$\displaystyle \frac{3 \sqrt{2}}{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{18}}{3}+\displaystyle \frac{6}{\sqrt{3}}$

process
$\displaystyle \frac{3 \sqrt{2}}{3}+\displaystyle \frac{6\sqrt{3}}{3}$

$\sqrt{2}+2\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{10\sqrt{2}-\sqrt{50}}{\sqrt{10}}$

process
$\displaystyle \frac{10 \times 2 \sqrt{5}-10\sqrt{5}}{10}$

$\sqrt{5}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{5}{\sqrt{8}}-\displaystyle \frac{3\sqrt{6}}{\sqrt{2}}$

process
$\displaystyle \frac{5}{2\sqrt{2}}-3\sqrt{3}=\displaystyle \frac{5\sqrt{2}}{2\times 2}-3\sqrt{3}$

$\displaystyle \frac{5\sqrt{2}}{4}-3\sqrt{3}$

次の分数の分母を有理化せよ。\\
$\displaystyle \frac{6}{\sqrt{32}}$

process
与式$=\displaystyle \frac{6}{4\sqrt{2}}
=\displaystyle \frac{6\sqrt{2}}{4\sqrt{2} \times \sqrt{2}}$

$\displaystyle \frac{3\sqrt{2}}{4}$











[Level7]
分母の有理化（+-を使う）
次の式を簡単にしなさい。\\
$\displaystyle\frac{2}{2+\sqrt{5}}$

process
$\displaystyle\frac{2(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})}=\displaystyle\frac{4-2\sqrt{5}}{-1}=-4+2\sqrt{5}$

$-4+2\sqrt{5}$

次の式を簡単にしなさい。\\
$\sqrt{6}-\displaystyle \frac{\sqrt{3}+\sqrt{2}}{\sqrt{2}}$

process
$\sqrt{6}-\displaystyle \frac{\sqrt{6}+2}{2}$

$\displaystyle \frac{\sqrt{6}}{2}-1$

次の式を簡単にしなさい。\\
$\displaystyle \frac{1}{\sqrt{5}-\sqrt{3}}+\displaystyle \frac{1}{\sqrt{5}+\sqrt{3}}$

process
$\displaystyle \frac{(\sqrt{5}+\sqrt{3})+(\sqrt{5}-\sqrt{3})}{5-3}$

$\sqrt{5}$

次の式を簡単にしなさい。\\
$\displaystyle \frac{2+\sqrt{3}}{2-\sqrt{3}}$

process
$\displaystyle  \frac{(2+\sqrt{3})^2}{(2-\sqrt{3})(2+\sqrt{3})}$

$7+4\sqrt{3}$

次の式を簡単にしなさい。\\
$\displaystyle \frac{1}{\sqrt{5}-2}+\displaystyle \frac{1}{\sqrt{5}+2}$

process
$\displaystyle \frac{\sqrt{5}+2+\sqrt{5}-2}{(\sqrt{5})^2-2^2}$

$2\sqrt{5}$

次の式を有理化しなさい。\\
$\displaystyle \frac{\sqrt{2}}{2-\sqrt{2}}$

process
$\displaystyle \frac{\sqrt{2} (2+\sqrt{2})}{(2-\sqrt{2})(2+\sqrt{2})}$

$\sqrt{2} +1$

次の式を簡単にしなさい。\\
$\sqrt{32}+\displaystyle \frac{\sqrt{2}+1}{\sqrt{2}-1}$

process
$4 \sqrt{2}+\displaystyle \frac{(\sqrt{2}+1)^2}{(\sqrt{2}-1)(\sqrt{2}+1)}$

$3+6\sqrt{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{1}{\sqrt{3}+1}$

$\displaystyle \frac{\sqrt{3}-1}{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{5}}{\sqrt{5}-2}$

$5+2 \sqrt{5}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{6}-2}{\sqrt{6}+2}$

$5-2 \sqrt{6}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}$

$\displaystyle \frac{5+\sqrt{21}}{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{4}{\sqrt{7}-\sqrt{3}}$

$\sqrt{7}+\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{7}}{\sqrt{7}+\sqrt{3}}$

$\displaystyle \frac{7-\sqrt{21}}{4}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{5}-2}{\sqrt{5}+2}$

$9-4 \sqrt{5}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

$4+ \sqrt{15}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{2}{\sqrt{6}+2}$

$\sqrt{6}-2$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{7}-2}{\sqrt{7}+2}$

$\displaystyle \frac{11-4\sqrt{7}}{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{1}{\sqrt{3}+1}+\displaystyle \frac{1}{\sqrt{3}-1}$

$\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}+1}+\displaystyle \frac{\sqrt{3}+1}{\sqrt{3}-1}$

$4$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\displaystyle \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

$8$

次の式を簡単にしなさい。\\
$\displaystyle \frac{\sqrt{5}+2}{\sqrt{5}-2}-\sqrt{45}$

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

$9+\sqrt{5}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{1}{3-\sqrt{2}}$

process
$\displaystyle \frac{3+\sqrt{2}}{(3-\sqrt{2})(3+\sqrt{2})}\\
=\displaystyle \frac{3+\sqrt{2}}{7}$

$\displaystyle \frac{3+\sqrt{2}}{7}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{1}{2-\sqrt{3}}$

process
$\displaystyle \frac{1 \times (2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}
=2+\sqrt{3}$

$2+\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{\sqrt{2}}{2-\sqrt{2}}$

process
$\displaystyle \frac{\sqrt{2}}{2-\sqrt{2}}
=\displaystyle \frac{\sqrt{2}(2+\sqrt{2})}{(2-\sqrt{2})(2+\sqrt{2})}\\
=\displaystyle \frac{2+2\sqrt{2}}{2}=1+\sqrt{2}$

$1+\sqrt{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{4}{\sqrt{7}+\sqrt{3}}$

process
$\displaystyle \frac{4(\sqrt{7}-\sqrt{3})}{(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})}
=\displaystyle \frac{4(\sqrt{7}-\sqrt{3})}{4}$

$\sqrt{7}-\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{1}{\sqrt{7}+1}$

process
$\displaystyle \frac{1}{\sqrt{7}+1}
=\displaystyle \frac{1 \times (\sqrt{7}-1)}{(\sqrt{7}+1)(\sqrt{7}-1)}
=\displaystyle \frac{\sqrt{7}-1}{7-1}$

$\displaystyle \frac{\sqrt{7}-1}{6}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{8}{\sqrt{5}+1}$

process
$\displaystyle \frac{8(\sqrt{5}-1)}{(\sqrt{5}+1)(\sqrt{5}-1)}
=\displaystyle \frac{8(\sqrt{5}-1)}{5-1}\\=2(\sqrt{5}-1)$

$2\sqrt{5}-2$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{1}{\sqrt{6}-2}$

process
$\displaystyle\frac{\sqrt{6}+2}{6-4}$

$\displaystyle\frac{\sqrt{6}+2}{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{2}{\sqrt{3}+1}$

process
$\displaystyle\frac{2\left( \sqrt{3}-1\right)}{3-1}$

$\sqrt{3}-1$

次の式を有理化しなさい。\\
$\displaystyle\frac{1}{\sqrt{7}+\sqrt{5}}$

process
$\displaystyle\frac{\sqrt{7}-\sqrt{5}}{(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})}$

$\displaystyle\frac{\sqrt{7}-\sqrt{5}}{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{2}{2-\sqrt{2}}$

$2+\sqrt{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{2}{\sqrt{5}-\sqrt{3}}$

$\sqrt{5}+\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{2}{\sqrt{5}-1}$

process
$\displaystyle\frac{2(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)}=\displaystyle\frac{\sqrt{5}+1}{2}$

$\displaystyle\frac{\sqrt{5}+1}{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{2}-1}$

process
$\left(\sqrt{2}-1\right)+\left(\sqrt{2}+1\right)$

$2\sqrt{2}$

次の式の分母を有理化しなさい。\\
$\displaystyle\frac{1}{2+\sqrt{2}}$

process
$\displaystyle\frac{1\times(2-\sqrt{2})}{(2+\sqrt{2})(2-\sqrt{2})}=\displaystyle\frac{2-\sqrt{2}}{4-2}$

$\displaystyle\frac{2-\sqrt{2}}{2}$

次の分数の分母を有理化せよ。\\
$\displaystyle \frac{1}{\sqrt{3}-\sqrt{2}}$

$\sqrt{3}+\sqrt{2}$

次の分数の分母を有理化せよ。\\
$\displaystyle \frac{7}{3+\sqrt{2}}$

process
$\displaystyle \frac{7(3-\sqrt{2})}{(3+\sqrt{2})(3-\sqrt{2})}$

$3-\sqrt{2}$

次の式を簡単にしなさい。\\
$\displaystyle\frac{13}{4-\sqrt{3}}$

$4+\sqrt{3}$

次の式の分母を有理化しなさい。\\
$\displaystyle \frac{2}{2-\sqrt{3}}$

$4+2\sqrt{3}$

$\displaystyle\frac{1}{\sqrt{7}+\sqrt{6}}$は，分母を有理化すると，
$\sqrt{\fbox{\bf\, エ \,}}-\sqrt{\fbox{\bf\, オ \,}}$になる。

process
$\displaystyle\frac{1}{\sqrt{7}+\sqrt{6}}=\sqrt{7}-\sqrt{6}$

$\sqrt{7}-\sqrt{6}$

次の式を簡単にしなさい。\\
$\displaystyle\frac{\sqrt{3}}{2-\sqrt{3}}$

$3+2\sqrt{3}$

次の式を簡単にしなさい。\\
$\displaystyle\frac{3\sqrt{5}}{\sqrt{5}-\sqrt{2}}$

$5+\sqrt{10}$

次の式を簡単にしなさい。\\
$\displaystyle\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}$

$3+\sqrt{6}$






[Level8]


[EOF]