Determine the value of ‘x’ in \(x{\rm{}} = {\rm{}}\frac{1}{{1{\rm{\;}} + {\rm{\;}}\sqrt 2 }}{\rm{}} + {\rm{}}\frac{1}{{\sqrt 2 {\rm{\;}} + {\rm{\;}}\sqrt 3 }}{\rm{}} + {\rm{}}\frac{1}{{\sqrt 3 {\rm{\;}} + {\rm{\;}}2}}\)
Determine the value of ‘x’ in \(x{\rm{}} = {\rm{}}\frac{1}{{1{\rm{\;}} + {\rm{\;}}\sqrt 2 }}{\rm{}} + {\rm{}}\frac{1}{{\sqrt 2 {\rm{\;}} + {\rm{\;}}\sqrt 3 }}{\rm{}} + {\rm{}}\frac{1}{{\sqrt 3 {\rm{\;}} + {\rm{\;}}2}}\)
1). -1
2). 0
3). 1
4). 2
1 answers
$(x = \frac{1}{{1\; + \;\sqrt 2 }} \times \frac{{1\; - \;\sqrt 2 }}{{1\; - \;\sqrt 2 }} + \frac{1}{{\sqrt 2 \; + \;\sqrt 3 }} \times \frac{{\sqrt 2 \; - \;\sqrt 3 }}{{\sqrt 2 \; - \;\sqrt 3 }} + \frac{1}{{\sqrt 3 \; + \;2}} \times \frac{{\sqrt 3 \; - \;2}}{{\sqrt 3 \; - \;2}})$
$( \Rightarrow x = \frac{{1\; - \;\sqrt 2 }}{{1^2 - \left( {\sqrt 2} \right)^2}} + \frac{{\sqrt 2 - \;\sqrt 3 }}{{\left( {\sqrt 2} \right)^2 - \left( {\sqrt 3} \right)^2}} + \frac{{\sqrt 3 - \;2}}{{\left( {\sqrt 3} \right)^2\; - \;2^2}})$
$(\Rightarrow x = \frac{{1\; - \;\sqrt 2 }}{{1\; - \;2}} + \frac{{\sqrt 2 \; - \;\sqrt 3 }}{{2\; - \;3}} + \frac{{\sqrt 3 \; - \;2}}{{3\; - \;4}})$
$(\Rightarrow x = -\;\frac{{\left( {1\; - \;\sqrt 2 } \right)}}{1}-\;\frac{{\left( {\sqrt 2 - \sqrt 3 } \right)}}{1}-\;\frac{{\left( {\sqrt 3 - 2} \right)}}{1})$
⇒ x = –1 + √2 – √2 + √3– √3 + 2
∴ x = 1