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A⁵ + b⁵ = (a + b)(a⁴ – a³b + a²b² – ab³ + b⁴)

= (a + b)(a⁴ + b⁴ + a²b² – ab³ – a³b )

= (a + b)(a⁴ + b⁴ + a²b² – ab (a² + b²))

a = x

b = 1/x

x² + 1/x² = 5 (given)

(x+1/x)² = x² + 1/x² + 2*x*1/x

(x+1/x)² = 5 + 2

(x+1/x)² = 7

x+1/x = √7

(x² + 1/x²)² = x⁴ + 1/x⁴ + 2*x²*(1/x²)

5² = x⁴ + 1/x⁴ + 2

25 - 2 =x⁴ + 1/x⁴

x⁴ + 1/x⁴ = 23

from formula;

x⁵ + 1/x⁵ = (x + 1/x)[(x⁴ + 1/x⁴) + (x² * 1/x²) -(x * 1/x)(x² + 1/x²) )

= (√7) [ (23) + (1) - (1)(5)]

= √7 ( 24 -5)

=

= (a + b)(a⁴ + b⁴ + a²b² – ab³ – a³b )

= (a + b)(a⁴ + b⁴ + a²b² – ab (a² + b²))

a = x

b = 1/x

x² + 1/x² = 5 (given)

(x+1/x)² = x² + 1/x² + 2*x*1/x

(x+1/x)² = 5 + 2

(x+1/x)² = 7

x+1/x = √7

(x² + 1/x²)² = x⁴ + 1/x⁴ + 2*x²*(1/x²)

5² = x⁴ + 1/x⁴ + 2

25 - 2 =x⁴ + 1/x⁴

x⁴ + 1/x⁴ = 23

from formula;

x⁵ + 1/x⁵ = (x + 1/x)[(x⁴ + 1/x⁴) + (x² * 1/x²) -(x * 1/x)(x² + 1/x²) )

= (√7) [ (23) + (1) - (1)(5)]

= √7 ( 24 -5)

=

**17√7**