Question

If x1/4 + x-1/4 = 2, then what is the value of x81 + (1/x81)?

Answer 1

I think your question is if x¼ + 1/x¼ =2 then, what is the value of x⁸¹ + 1/x⁸¹= ?

x¼ + 1/x¼ = 2
squaring both sides,
=> (x¼ + 1/x¼)² = 2² = 4
=> x½ + 1/x½ + 2.x½.1/x½ = 4
=> x½ + 1/x½ + 2 = 4
=> x½ + 1/x½ = 2
again, squaring both sides,
x + 1/x +2.x½.1/x½ =4
=> x + 1/x = 2
now take cube both sides,
x³ + 1/x³ +3x.1/x(x + 1/x) = 8
=> x³ + 1/x³ = 8 - 6 = 2
so, x³ⁿ + 1/x³ⁿ = 2
here, x⁸¹ = x³⁽²⁷⁾
hence, x⁸¹ + 1/x⁸¹ = 2

2nd method //
if two positive real number a and 1/a are given
their sum = 2
then,you can assume a = 1 and b = 1
like here , x¼ + 1/x¼ = 2 ,
if we let x¼ = a
then, a + 1/a = 2
=> a² - 2a + 1= 0
=> a = 1 = x¼ so, x = 1
hence, x⁸¹ = (1)⁸¹ = 1 and 1/x⁸¹ = 1
so, x⁸¹ + 1/x⁸¹ = 2

Answer 2

find the value of (-1)²⁷.