Question

Thực hiện phép tính:

a. \(\dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+...+\dfrac{1}{\sqrt{99}+\sqrt{100}}\)

b. \(\left(\dfrac{2}{\sqrt{3}-1}+\dfrac{3}{\sqrt{3}-2}+\dfrac{15}{3-\sqrt{3}}\right)\times\dfrac{1}{\sqrt{3}+5}\)

Answer 1

`a)1/[1+\sqrt{2}]+1/[\sqrt{2}+\sqrt{3}]+...+1/[\sqrt{99}+\sqrt{100}]`

`=[1-\sqrt{2}]/[1-2]+[\sqrt{2}-\sqrt{3}]/[2-3]+...+[\sqrt{99}-\sqrt{100}]/[99-100]`

`=-(1-\sqrt{2}+\sqrt{2}-\sqrt{3}+...+\sqrt{99}-\sqrt{100})`

`=-(1-\sqrt{100})=\sqrt{100}-1`

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`b)(2/[\sqrt{3}-1]+3/[\sqrt{3}-2]+15/[3-\sqrt{3}]). 1/[\sqrt{3}+5]`

=([2(\sqrt{3}+1)]/[3-1]+[3(\sqrt{3}+2)]/[3-4]+[5\sqrt{3}(\sqrt{3}+1)]/[3-1]). 1/[\sqrt{3}+5]`

`=(\sqrt{3}+1-3\sqrt{3}-6+[15+5\sqrt{3}]/2). 1/[\sqrt{3}+5]`

`=[-4\sqrt{3}-10+15+5\sqrt{3}]/2 . 1/[\sqrt{3}+5]`

`=[5+\sqrt{3}]/2 . 1/[\sqrt{3}+5]=1/2`