Question

the value of √1+2008√1+2009√1+2010√1+2011*2013
i need and understandable solution

Answer 1

Answer:

It's 2009

Step-by-step explanation:

We know that : ( x + 1 )( x - 1 ) = x^2 -1,

so, root of 1 + (2011)(2013) = root of 1 + 2012^2 - 1 = 2012 itself,

with this following pattern , we can arrive at the answer with 2009.

hence solved xD

Answer 2

$\displaystyle \sf{ \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + 2011 \times 2013} } } } ))) } = \bf 2009$

Given :

$\displaystyle \sf{ \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + 2011 \times 2013} } } } ))) }$

To find :

Simplify the expression

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\displaystyle \sf{ \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + 2011 \times 2013} } } } ))) }$

Step 2 of 2 :

Simplify the given expression

$\displaystyle \sf{ \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + 2011 \times 2013} } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + \{2012 - 1 \} \times \{2012 + 1 \}} } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + \{ {2012}^{2} - {1}^{2} \} } } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{1 + \{ {2012}^{2} - 1 \} } } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \sqrt{ \{ {2012}^{2} \} } } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + 2010 \times 2012 } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + \{2011 - 1 \} \times \{2011 + 1 \} } } } ))) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + \{ {2011}^{2} - {1}^{2} \} } } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{(1 + \{ {2011}^{2} - 1 \} } } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \sqrt{( \{ {2011}^{2} \} } } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + 2009 \times 2011 } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + \{2010 - 1 \} \times \{2010 + 1 \} } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + \{ {2010}^{2} - {1}^{2} \} } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{(1 + {2010}^{2} -1} } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \sqrt{( {2010}^{2} } } )) }$

$\displaystyle \sf{ = \sqrt{(1 + 2008 \times 2010} ) }$

$\displaystyle \sf{ = \sqrt{(1 + \{2009 - 1 \} \times \{2009 + 1 \}} ) }$

$\displaystyle \sf{ = \sqrt{(1 + \{ {2009}^{2} - {1}^{2} \} } ) }$

$\displaystyle \sf{ = \sqrt{(1 +{2009}^{2} - 1} ) }$

$\displaystyle \sf{ = \sqrt{{2009}^{2} } }$

$\displaystyle \sf{ = 2009}$

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