problem solving

 As a mixed power relation show that  (x)⁵ + (x)⁵ = 2(x)⁵ can be expressed  as a power with exponent other thn 5 or multiples of 5

Let x = 2ⁿ k^{m ,} then  (2ⁿ k^m)⁵ + (2ⁿ k^m)⁵ = 2(2ⁿ k^m)⁵ = 2⁵ⁿ⁺¹ k^5m

when n = 1 ,m=2 ; (2 k²)⁵ + (2 k²)⁵ = 2(2 k²)⁵ = (2³ k⁵)²

Keeping n same and m is changed

when n = 1 ,m=4 ; (2 k⁴)⁵ + (2 k⁴)⁵ = 2(2 k⁴)⁵ = (2³ k¹⁰)²

Keeping m is same and n is changed

when n = 3 ,m=2 ; (2³ k²)⁵ + (2³ k²)⁵ = 2(2³ k²)⁵ = (2⁸ k⁵)²

when n=1 m=3 ; (2 k³)⁵ + (2 k³)⁵ = 2(2 k³)⁵ = (2² k⁵)³

when n=3 m=4 ; (2³ k⁴)⁵ + (2³ k⁴)⁵ = 2(2³ k⁴)⁵ = (2⁴ k⁵)⁴

when n=1 m=6 ; (2 k⁶)⁵ + (2 k⁶)⁵ = 2(2 k⁶)⁵ = (2 k⁵)⁶

In 2⁵ⁿ⁺¹ k^5m the exponents 5n+1 cannot be expressed as a multiple of 5. Hence sum of two power number with exponent 5 cannot be expressed as a number with same exponent. This is in accordance with FLT.