The simplest value of \(\frac{1}{{\sqrt 2+ \sqrt 3 }} + \frac{1}{{\sqr

1.	The simplest value of \(\frac{1}{{\sqrt 2+ \sqrt 3 }} + \frac{1}{{\sqrt 3+ \sqrt 4 }} + \frac{1}{{\sqrt 4+ \sqrt 5 }} + \frac{1}{{\sqrt 5+ \sqrt 6 }}\) is1). √3 (√2-1)2). √2 (√3-1)3). √3-14). √2-1
Answer» $(\FRAC{1}{{\sqrt 2+ \sqrt 3 }} + \frac{1}{{\sqrt 3+ \sqrt 4 }} + \frac{1}{{\sqrt 4+ \sqrt 5 }} + \frac{1}{{\sqrt 5+ \sqrt 6 }})$ $(= \frac{{(\sqrt 2- \sqrt {3)} }}{{\left( {\sqrt 2+ \sqrt 3 } \right)(\sqrt 2- \sqrt {3)} }} + \frac{{\left( {\sqrt 3- \sqrt 4 } \right)}}{{\left( {\sqrt 3+ \sqrt 4 } \right)\left( {\sqrt 3- \sqrt 4 } \right)}} + \frac{{(\sqrt 4- \sqrt {5)} }}{{(\sqrt 4+ \sqrt {5)} (\sqrt 4- \sqrt {5)} }} + \frac{{(\sqrt 5- \sqrt {6)} }}{{(\sqrt 5+ \sqrt {6)} (\sqrt 5- \sqrt {6)} }})$ [Dividing the NUMERATOR & DENOMINATOR of each fraction by same term] $(= \frac{{(\sqrt 2- \sqrt {3)} }}{{{{\sqrt 2 }^2} - {{\sqrt 3 }^2}}} + \frac{{\left( {\sqrt 3- \sqrt 4 } \right)}}{{{{\sqrt 3 }^2} - {{\sqrt 4 }^2}}} + \frac{{(\sqrt 4- \sqrt {5)} }}{{{{\sqrt 4 }^2} - {{\sqrt 5 }^2}}} + \frac{{(\sqrt 5- \sqrt {6)} }}{{{{\sqrt 5 }^2} - {{\sqrt 6 }^2}}})$ $(= \frac{{(\sqrt 2- \sqrt {3)} }}{{ - 1}} + \frac{{\left( {\sqrt 3- \sqrt 4 } \right)}}{{ - 1}} + \frac{{(\sqrt 4- \sqrt {5)} }}{{ - 1}} + \frac{{(\sqrt 5- \sqrt {6)} }}{{ - 1}})$ = - (√2 - √3) - (√3 - √4) - (√4 - √5) - (√5 - √6) = √3 - √2 + √4 -√3 + √5 - √4 + √6 - √5 = √6 - √2 = √3 × √2 - √2 = √2 [√3 – 1]

The simplest value of $\frac{1}{{\sqrt 2+ \sqrt 3 }} + \frac{1}{{\sqrt 3+ \sqrt 4 }} + \frac{1}{{\sqrt 4+ \sqrt 5 }} + \frac{1}{{\sqrt 5+ \sqrt 6 }}$ is1). √3 (√2-1)2). √2 (√3-1)3). √3-14). √2-1

Answer»

$(\FRAC{1}{{\sqrt 2+ \sqrt 3 }} + \frac{1}{{\sqrt 3+ \sqrt 4 }} + \frac{1}{{\sqrt 4+ \sqrt 5 }} + \frac{1}{{\sqrt 5+ \sqrt 6 }})$

$(= \frac{{(\sqrt 2- \sqrt {3)} }}{{\left( {\sqrt 2+ \sqrt 3 } \right)(\sqrt 2- \sqrt {3)} }} + \frac{{\left( {\sqrt 3- \sqrt 4 } \right)}}{{\left( {\sqrt 3+ \sqrt 4 } \right)\left( {\sqrt 3- \sqrt 4 } \right)}} + \frac{{(\sqrt 4- \sqrt {5)} }}{{(\sqrt 4+ \sqrt {5)} (\sqrt 4- \sqrt {5)} }} + \frac{{(\sqrt 5- \sqrt {6)} }}{{(\sqrt 5+ \sqrt {6)} (\sqrt 5- \sqrt {6)} }})$

[Dividing the NUMERATOR & DENOMINATOR of each fraction by same term]

$(= \frac{{(\sqrt 2- \sqrt {3)} }}{{{{\sqrt 2 }^2} - {{\sqrt 3 }^2}}} + \frac{{\left( {\sqrt 3- \sqrt 4 } \right)}}{{{{\sqrt 3 }^2} - {{\sqrt 4 }^2}}} + \frac{{(\sqrt 4- \sqrt {5)} }}{{{{\sqrt 4 }^2} - {{\sqrt 5 }^2}}} + \frac{{(\sqrt 5- \sqrt {6)} }}{{{{\sqrt 5 }^2} - {{\sqrt 6 }^2}}})$

$(= \frac{{(\sqrt 2- \sqrt {3)} }}{{ - 1}} + \frac{{\left( {\sqrt 3- \sqrt 4 } \right)}}{{ - 1}} + \frac{{(\sqrt 4- \sqrt {5)} }}{{ - 1}} + \frac{{(\sqrt 5- \sqrt {6)} }}{{ - 1}})$

= - (√2 - √3) - (√3 - √4) - (√4 - √5) - (√5 - √6)

= √3 - √2 + √4 -√3 + √5 - √4 + √6 - √5

= √6 - √2

= √3 × √2 - √2

= √2 [√3 – 1]

The simplest value of \(\frac{1}{{\sqrt 2+ \sqrt 3 }} + \frac{1}{{\sqrt 3+ \sqrt 4 }} + \frac{1}{{\sqrt 4+ \sqrt 5 }} + \frac{1}{{\sqrt 5+ \sqrt 6 }}\) is1). √3 (√2-1)2). √2 (√3-1)3). √3-14). √2-1

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