Question

$a + \frac{1}{a} = \sqrt{3} \: then \: value \: of \: a {6}^{} - \frac{1}{a {6}^{} } \ + 2 =$

Answer 1

a + 1/a = √3
a³ + 1/a³ = (a + 1/a)³ - 3a.1/a(a + 1/a)
= (a + 1/a)³ - 3(a + 1/a)
=(√3)³ - 3(√3)
= 3√3 - 3√3 = 0

e.g., a³ + 1/a³ = 0
a^6 + 1 = 0
a^6 = -1
so, 1/a^6 = -1

now,
a^6 - 1/a^6 + 2
= -1 - (-1) + 2
= -1 + 1 + 2 = 2