Question

If (x^4) - (1/x^4)=816√2, what is x???????

Answer 1

x = ±∜(577+408√2)

and six complex roots
No useful tricks on something like this. Do note that
x^4 + 1/x^4 = (x^2 + 1/x^2)^2 - 2
= (x^2 + 1/x^2 + √2)(x^2 + 1/x^2 - √2)
and since x^2 + 1/x^2 = (x + 1/x)^2 - 2
that takes you nearer to the final solution
It would have been nice had the final answer been a nice rational number.
If it had been (x^4) - (1/x^4) = 0
the roots are ±1, ±i, ±(1±i)/2

Answer 2

After scribbling on about 3 sheets of paper, and getting nowhere, I tried the obvious

x^4 - 1/x^4 = 816√2
multiply by x^4

x^8 - 1 = 816√2 x^4
x^8 - 816√2 x^4 + ...... = 1 + ....... , now complete the square

x^8 - 816√2x^4 + 332928 = 332929
(x^4 - 408√2)^2 = √332929 = 577
x^4 - 408√2 = 577
x^4 = 577 + 408√2
x = ± (577 + 408√2)^(1/4) , which is the real number given by Wolfram