Is there a solution to 16^2-8x+1=0

Yes X = 32, here is the procedure

If ya meant 16x²-8x+1=0 then

complete square

group x

(16x²-8x) + 1=0

undistribute 16

16 (x²-0.5x) + 1=0

take 1/2 of linear coefient and squre positive and negaitve of it inside parentheasees

-0.5/2=-0.25, (-0.25) ^2=0.0625

16 (x²-0.5x+0.0625-0.0625) + 1=0

factor perfect square trinomial

16 ((x-0.25) ²-0.0625) + 1=0

expand

16 (x-0.25) ²-1+1=0

16 (x-0.25) ²=0

(x-0.25) ²=0

x-0.25=0

x=0.25

or, if you mean

16²-8x+1=0

256-8x+1=0

257-8x=0

add 8x both sides

257=8x

divide both sides by 8

257/8=x