Question

1 upon 2 + root 5 + 1 upon root 5 + root 6 + 1 upon root 6 + root 7 + 1 upon root 7 + root 8

Answer 1

If i am wrong, then please correct me.

Answer 2

Answer:

$\frac{1}{2+\sqrt{5} } + \frac{1}{\sqrt{5}+\sqrt{6} } + \frac{1}{\sqrt{6}+\sqrt{7} } + \frac{1}{\sqrt{7}+\sqrt{8} }$ = 2(√2 - 1)

Step-by-step explanation:

We have to simplify the expression: $\frac{1}{2+\sqrt{5} } + \frac{1}{\sqrt{5}+\sqrt{6} } + \frac{1}{\sqrt{6}+\sqrt{7} } + \frac{1}{\sqrt{7}+\sqrt{8} }$

At 1st we will simplify each term, then add them.

• Simplification of  $\frac{1}{2+\sqrt{5} }$:

         $\frac{1}{2+\sqrt{5} }$

       = $\frac{1*(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5}) }$ [ By multiplying (2-√5) to nominator and de-nominator]

        = $\frac{2-\sqrt{5} }{2^{2}-\sqrt{5} ^{2} }$            [as (a+b)(a-b) = a²-b²]

        = - (2- √5) = -2+√5

• Simplification of  $\frac{1}{\sqrt{5}+\sqrt{6} }$:

       $\frac{1}{\sqrt{5}+\sqrt{6} }$

  = $\frac{\sqrt{5}-\sqrt{6} }{(\sqrt{5}+\sqrt{6})(\sqrt{5}-\sqrt{6} ) }$ [By multiplying (√5-√6)to nominator and de-nominator]

    = $\frac{\sqrt{5}-\sqrt{6} }{\sqrt{5} ^{2} - \sqrt{6} ^{2} }$            [as (a+b)(a-b) = a²-b²]

    = - (√5-√6) = -√5+√6

• Simplification of  $\frac{1}{\sqrt{6}+\sqrt{7} }$:

        $\frac{1}{\sqrt{6}+\sqrt{7} }$

  = $\frac{\sqrt{6}-\sqrt{7} }{(\sqrt{6}+\sqrt{7})(\sqrt{6}-\sqrt{7} ) }$ [By multiplying (√6-√7)to nominator and de-nominator]

    = $\frac{\sqrt{6}-\sqrt{7} }{\sqrt{6} ^{2} - \sqrt{7} ^{2} }$            [as (a+b)(a-b) = a²-b²]

    = - (√6-√7) = -√6+√7

• Simplification of  $\frac{1}{\sqrt{7}+\sqrt{8} }$:

       $\frac{1}{\sqrt{7}+\sqrt{8} }$

  = $\frac{\sqrt{7}-\sqrt{8} }{(\sqrt{7}+\sqrt{8})(\sqrt{7}-\sqrt{8} ) }$ [By multiplying (√7-√8)to nominator and de-nominator]

    = $\frac{\sqrt{7}-\sqrt{8} }{\sqrt{7} ^{2} - \sqrt{8} ^{2} }$            [as (a+b)(a-b) = a²-b²]

    = - (√7-√8) = -√7+√8

• Addition:

                       $\frac{1}{2+\sqrt{5} } + \frac{1}{\sqrt{5}+\sqrt{6} } + \frac{1}{\sqrt{6}+\sqrt{7} } + \frac{1}{\sqrt{7}+\sqrt{8} }$

                   = (-2+√5) + (-√5+√6) + (-√6+√7) + (-√7+√8)

                   = -2+√5-√5+√6-√6+√7-√7+√8

                   = -2 +√8

                   = -2 + 2√2

                   = 2(√2 - 1)