Question

If
`\sf x^((x^4)) = 4,` then find the value of
`\sf x^((x^2)) + x^((x^8)).`​

Answer 1

By inspection, we can see that x = ±√2, since (±√2)⁴ = 2² = 4.

In particular, if
`x^n=n`, which means
`x=n^(\frac1n)`, then
`x^(x^n)=x^n=n`.

Let x = √2. Then

`x^(x^2)+x^(x^8)=(\sqrt2)^((\sqrt2)^2)+(\sqrt2)^((\sqrt2)^8)`

`x^(x^2)+x^(x^8)=(\sqrt2)^2+(\sqrt2)^(16)`

`x^(x^2)+x^(x^8)=2+256`

`x^(x^2)+x^(x^8)=\boxed{258}`