Question

[tex]if a=5+ \sqrt[2]{6} and b=1/a then what willbe the value of a^{2} + b ^{2} and

Answer 1

a = 5 + 2√6

b = 1/a = 1/( 5 +2√6) = (5-2√6)/(25-24)
= ( 5 -2√6)

1/a = 5 -2√6

now,

a² + b² = a² + 1/a² = (a +1/a)² -2.a .1/a
= (a + 1/a)² -2

=(5 +2√6 +5 -2√6)² -2

=(10)² -2

=98 ( answer )

Answer 2

$if\ a=5+ \sqrt[2]{6}\ and\ b=1/a\ then\ what\ will be\ the\ value\ of\ a^{2} + b^{2}$
=====   Is this the question ??   
a =  5 +  √6
b = 1/a = 1/(5 +√6)
    = (5-√6)/[(5-√6)(5+√6)]    rationalization of denominator.
    = (5 - √6) / [25 - 6]
    = (5 - √6) /19

a² + b² = 5² + 6 + 10√6  +  [5² + 6 - 10 √6] /361
           = [11222 + 3600 √6] /361

=========================
$if\ a=5+ 2\sqrt{6}\ and\ b=1/a\ then\ what\ will be\ the\ value\ of\ a^{2} + b^{2}$
=====   Is this the question ??   
a =  5 + 2√6 
b = 1/a = 1/(5 +2 √6)
    = (5-2√6) / [(5-2√6)(5+2√6)]    rationalization of denominator.
    = (5 - 2√6) / [25 - 4*6]
    = (5 - 2√6) 

a² + b² = [5² + 4*6 + 20√6]  +  [5² + 4*6 - 20 √6] 
           = 98