1
\(=\dfrac{15.\sqrt{20}}{3\sqrt{20}.\sqrt{20}}=\dfrac{15\sqrt{20}}{3.20}=\dfrac{15\sqrt{20}}{60}=\dfrac{\sqrt{15^2.20}}{60}=\dfrac{\sqrt{4500}}{60}=\dfrac{30\sqrt{5}}{60}=\dfrac{\sqrt{5}}{2}\)
2
\(=\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}=\sqrt{3}+1\)
3
\(=\dfrac{1\left(\sqrt{2}+1\right)}{2-1}=\dfrac{\sqrt{2}+1}{1}=\sqrt{2}+1\)
4
\(=\dfrac{\left(\sqrt{15}-\sqrt{6}\right)\left(\sqrt{2}+\sqrt{5}\right)}{2-5}=\dfrac{\sqrt{30}+\sqrt{75}-\sqrt{12}-\sqrt{30}}{-3}=\dfrac{\sqrt{5^2.3}-\sqrt{2^2.3}}{-3}=\dfrac{5\sqrt{3}-2\sqrt{3}}{-3}=\dfrac{3\sqrt{3}}{-3}=-\sqrt{3}\)
5
\(=\dfrac{\sqrt{3}.\sqrt{2}\left(\sqrt{3}-\sqrt{2}\right)}{\sqrt{2}-\sqrt{3}}=-\dfrac{\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}{\sqrt{3}-\sqrt{2}}=-\sqrt{6}\)
6
\(=\dfrac{2+\sqrt{3}}{4-3}+\dfrac{2-\sqrt{3}}{4-3}=2+\sqrt{3}+2-\sqrt{3}=4\)
7
\(=\sqrt{2^2-\left(\sqrt{3}\right)^2}=\sqrt{4-3}=1\)
4: \(=\dfrac{\sqrt{3}\left(\sqrt{5}-\sqrt{2}\right)}{-\left(\sqrt{5}-\sqrt{2}\right)}=-\sqrt{3}\)
7: \(=\sqrt{4-3}=1\)
3: \(=\dfrac{\sqrt{2}+1}{2-1}=\sqrt{2}+1\)
2: \(=\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}=\sqrt{3}+1\)
1. \(\dfrac{15}{3\sqrt{20}}\)
\(=\dfrac{5}{\sqrt{20}}\)
\(=\dfrac{\sqrt{5}}{2}\)
2. \(\dfrac{3+\sqrt{3}}{\sqrt{3}}\)
\(=\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}\)
\(=\sqrt{3}+1\)
3. \(\dfrac{1}{\sqrt{2}-1}\)
\(=\dfrac{\sqrt{2}+1}{2-1}\)
\(=1+\sqrt{2}\)
4. \(\dfrac{\sqrt{15}-\sqrt{6}}{\sqrt{2}-\sqrt{5}}\)
\(=\dfrac{\sqrt{3}\left(\sqrt{5}-\sqrt{2}\right)}{-\left(\sqrt{5}-\sqrt{2}\right)}\)
\(=\dfrac{\sqrt{3}}{-1}=-\sqrt{3}\)
5. \(\dfrac{3\sqrt{2}-2\sqrt{3}}{\sqrt{2}-\sqrt{3}}\)
\(=\dfrac{\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}{-\left(\sqrt{3}-\sqrt{2}\right)}\)
\(=\dfrac{\sqrt{6}}{-1}=-\sqrt{6}\)
6. \(\dfrac{1}{2-\sqrt{3}}+\dfrac{1}{2+\sqrt{3}}\)
\(=\dfrac{\left(2+\sqrt{3}\right)+\left(2-\sqrt{3}\right)}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}\)
\(=\dfrac{2+\sqrt{3}+2-\sqrt{3}}{4-3}\)
\(=\dfrac{4}{1}=4\)
7. \(\sqrt{2+\sqrt{3}}\sqrt{2-\sqrt{3}}\)
\(=\sqrt{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)
\(=\sqrt{4-3}\)
\(=\sqrt{1}=1\)
