Question

If a= 2+root3 find a3 +1/a3 and a3-1/a3

Answer 1

HELLO DEAR,


given that:-


a = (2 + √3)

a³ = (2 + √3)³

⇒a³ = [8 + 3√3 + 6√3(2 + √3) ]

⇒a³ = (8 + 3√3 + 12√3 + 18)

⇒a³ = (26 + 15√3)-----------( 1 )


now,

1/a³ = 1/(26 + 15√3) * (26 - 15√3) / (26 - 15√3)

⇒1/a³ = (26 - 15√3)/(676 - 675)

⇒1/a³ = (26 - 15√3) ----------- ( 2 )



adding--------( 1 ) & ------( 2)



we get,


(a³ + 1/a³) = (26 + 15√3) + (26 - 15√3)

⇒(a³ - 1/a³) = 26 + 26 + 15√3 - 15√3

⇒(a³ + 1/a³) = 52






Now,


subtracting ----- ( 1 ) & -----( 2)


(a³ - 1/a³) = (26 + 15√3) - (26 - 15√3)

⇒(a³ - 1/a³) = 26 - 26 + 15√3 + 15√3

⇒(a³ - 1/a³) = 30√3




I HOPE ITS HELP YOU DEAR,
THANKS