Question

3. Rút gọn các biểu thức:
a) 1/(5 + 3sqrt(2)) + 1/(5 - 3sqrt(2))
b) 1/(sqrt(5) + 1) + 1/(sqrt(5) - 2) - 1/(3 - sqrt(5)) - sqrt(5)
c) sqrt(3/20) + sqrt(1/60) - 2sqrt(1/15)

Answer 1

a: \(\dfrac{1}{5+3\sqrt{2}}+\dfrac{1}{5-3\sqrt{2}}\)

\(=\dfrac{5-3\sqrt{2}+5+3\sqrt{2}}{\left(5+3\sqrt{2}\right)\left(5-3\sqrt{2}\right)}\)

\(=\dfrac{10}{25-18}=\dfrac{10}{7}\)

b: \(\dfrac{1}{\sqrt{5}+1}+\dfrac{1}{\sqrt{5}-2}-\dfrac{1}{3-\sqrt{5}}-\sqrt{5}\)

\(=\dfrac{\left(\sqrt{5}-1\right)}{5-1}+\dfrac{\sqrt{5}+2}{5-4}-\dfrac{\left(3+\sqrt{5}\right)}{9-5}-\sqrt{5}\)

\(=\dfrac{1}{4}\left(\sqrt{5}-1\right)+\sqrt{5}+2-\dfrac{1}{4}\left(3+\sqrt{5}\right)-\sqrt{5}\)

\(=\dfrac{1}{4}\left(\sqrt{5}-1-3-\sqrt{5}\right)+2\)

\(=\dfrac{1}{4}\cdot\left(-4\right)+2=2-1=1\)

c: \(\sqrt{\dfrac{3}{20}}+\sqrt{\dfrac{1}{60}}-2\sqrt{\dfrac{1}{15}}\)

\(=\sqrt{\dfrac{15}{100}}+\sqrt{\dfrac{15}{900}}-\dfrac{2\sqrt{15}}{15}\)

\(=\dfrac{1}{10}\sqrt{15}+\dfrac{1}{30}\sqrt{15}-\dfrac{2}{15}\sqrt{15}\)

\(=\sqrt{15}\left(\dfrac{1}{30}+\dfrac{1}{10}-\dfrac{2}{15}\right)\)

\(=\sqrt{15}\left(\dfrac{1}{30}+\dfrac{3}{30}-\dfrac{4}{30}\right)=0\)