Question

\(\dfrac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}\)- \(\dfrac{6+2\sqrt{6}}{\sqrt{3}+\sqrt{2}}\)

\(\dfrac{5\sqrt{7}-\sqrt{21}}{\sqrt{75}-3}\) + \(\dfrac{10-5\sqrt{5}}{\sqrt{12}-\sqrt{15}}\)

 

 

Answer 1

\(\dfrac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}-\dfrac{6+2\sqrt{6}}{\sqrt{3}+\sqrt{2}}\)

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

\(=\sqrt{3}-2\sqrt{3}=-\sqrt{3}\)

\(\dfrac{5\sqrt{7}-\sqrt{21}}{\sqrt{75}-3}+\dfrac{10-5\sqrt{5}}{\sqrt{12}-\sqrt{15}}\)

\(=\dfrac{\sqrt{7}\left(5-\sqrt{3}\right)}{\sqrt{3}\left(5-\sqrt{3}\right)}+\dfrac{5\left(2-\sqrt{5}\right)}{\sqrt{5}\left(2-\sqrt{5}\right)}\)

\(=\sqrt{\dfrac{7}{3}}+\sqrt{5}=\dfrac{\sqrt{21}}{3}+\sqrt{5}=\dfrac{\sqrt{21}+3\sqrt{5}}{3}\)

Answer 2

`(sqrt{15} - sqrt{12})/(sqrt{5} - 2) - (6+2sqrt{6})/(sqrt{3} + sqrt{2})`

`= (sqrt{3}(sqrt{5} - sqrt{4}))/(sqrt{5} - 2) - (sqrt{12}(sqrt{3} +sqrt{2}))/(sqrt{3} + sqrt{2})`

`= sqrt{3} - sqrt{12} `

`= sqrt{3} -2 sqrt{3} `

`= - sqrt{3}`

\(\dfrac{5\sqrt{7}-\sqrt{21}}{\sqrt{75}-3}+\dfrac{10-5\sqrt{5}}{\sqrt{12}-\sqrt{15}}\)

= \(\dfrac{\sqrt{7}\left(5-\sqrt{3}\right)}{\sqrt{3}\left(5-\sqrt{3}\right)}+\dfrac{5\left(2-\sqrt{5}\right)}{\sqrt{3}\left(2-\sqrt{5}\right)}\)

`= (sqrt{7})/sqrt{3} + 5/sqrt{3}`

`= (sqrt{7} + 5)/sqrt{3} `