If #x^(x^4) = 4#, then #x^(x^2) + x^(x^8)# will be equal to what?

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Answer 1 · 2016-11-16T07:45:05

#258#

In general equations of the form #x^(x^4) = a# are difficult to solve, but in the case of #x^(x^4) = 4# there is a guessable solution.

#sqrt(2)^(sqrt(2)^4) = sqrt(2)^4 = 4#

So #x = sqrt(2)# is one solution and #x = -sqrt(2)# also works.

Then we find:

#sqrt(2)^(sqrt(2)^2) + sqrt(2)^(sqrt(2)^8) = sqrt(2)^2 + sqrt(2)^16 = 2+256 = 258#

#color(white)()#
Footnote

Concerning the "guessing", notice that if:

#x^a = a#

then:

#x^(x^a) = x^a = a" "# too.

So in general, #x=root(a)(a)# is a solution of #x^(x^a) = a#