Question

if: x is not equal to 0 and x + 1/x=2; then show that: x²+1/x²=x³+1/x³=x⁴+1/x⁴​

Answer 1

x + 1/x = 2 ...(1)

Square on both sides :

=> (x + 1/x)² = 2²

=> (x)² + (1/x)² + 2(x × 1/x) = 4

=> x² + 1/x² + 2(1) = 4

=> x² + 1/x² = 2 ...(2)

Cube on both sides of (1) :

=> (x + 1/x)³ = 2³

=> x³ + 1/x³ + 3(x × 1/x)(x + 1/x) = 8

=> x³ + 1/x³ + 3(1)(2) = 8 {from (1)}

=> x³ + 1/x³ = 2 ...(3)

Square on both sides of x² + 1/x² :

=> (x² + 1/x²)² = 2²

=> (x²)² + (1/x²)² + 2(x² × 1/x²) = 4

=> x⁴ + 1/x⁴ + 2(1) = 4

=> x⁴ + 1/x⁴ = 2 ...(4)

From (1), (2), (3) and (4) :

x² + 1/x² = x³ + 1/x³ = x⁴ + 1/x⁴ = 2