Question

find the values of a and b in the following 5+2√3/7+4√3=a-b√3

Answer 1

LHS = (5 + 2√3 ) / ( 7 + 4√3 )

rationalize the denominator

= (5 + 2√3 ) ( 7 - 4√3 ) / [ ( 7 + 4√3 ) ( 7 - 4√3 ) 

= [ 5 ×7 - 5 × 4√3 + 2√3 × 7 - 2√3 × 4√3 ] / [ (7 )² - (4√3 )² ]

here we used ( x + y ) (x - y ) = x² - y²  identity

= [35 -20√3 + 14√3 -24 ] / [ 49 - 48 ]

= (11 - 6√3 )

therefore ,

LHS = RHS

11 - 6√3 = A - B√3

comparing both sides

A = 11,

B = 6