x^8-16=0.....?

x^8 = 16
( x ² ) ^4 = 16
x² = 2
x = ± √ 2

x^8 - 16 = 0
(x^4 + 4)(x^4 - 4) = 0
(x^4 + 4)(x^2 + 2)(x^2 - 2) = 0

x^4 + 4 = 0
x^4 = -4
x = √-4 (imaginary number)

x^2 + 2 = 0
x^2 = -2
x = √-2 (imaginary number)

x^2 - 2 = 0
x^2 = 2
x = √2
x = 1.41421356......

I agree with what intellectual did, but there should be one more solution

x^8 - 16 = 0
(x^4 + 4)(x^4 - 4) = 0
(x^4 + 4)(x^2 + 2)(x^2 - 2) = 0

x^4 + 4 = 0 <==no real solution

x^2 - 2 = 0
(x + √2 )(x -√2 ) = 0
x = √2 or x = - √2

There are eight eighth-roots:

x⁸ − 16 = 0
x⁸ = 16
x⁴ = ±4

Therefore:
x⁴ = 4 .... and .... x⁴ = −4
x² = ±2 .... and .... x² = ±2·i

Therefore:
x² = 2 .... and .... x² = −2 .... and ....x² = 2·i.... and .... x² = −2·i
x = ±√2 .. & .. x = ±i·√2 .. & .. x = ±√(2·i) .. & .. x = ±i·√(−2·i)

Solutions:
x = √2
x = −√2
x = i·√2
x = −i·√2
x = 1 + i
x = 1 − i
x = −1 + i
x = −1 − i


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One could also do this by factoring and finding roots...
x⁸ − 16 = 0
(x⁴ + 4)·(x⁴ − 4) = 0
(x⁴ + 4)·(x² + 2)·(x² − 2) = 0
(x⁴ + 4)·(x² + 2)·(x + √2)·(x + √2) = 0

You can continue factoring using imaginary terms:
(x² + 2·i)·(x² − 2·i)·(x² + 2)·(x + √2)·(x + √2) = 0
(x² + 2·i)·(x² − 2·i)·(x² + 2)·(x + √2)·(x + √2) = 0

etcetera...

====
... but I prefer the polar method:

x⁸ − 16 = 0
x⁸ = 16
x⁸ = 16 · cis(0° + 360°·n)
x = [ 16 · cis(360°·n) ]^(⅛)
x = 16^(⅛) · [ cis (360°·n) ]^(⅛)
x = √2 · cis [ ⅛ ·(360°·n) ]
x = √2 · cis (45°·n)

we shall do this by factoring, not by bringin 16 to right side directly.
x^8-16=0
(x^4+4)(x^4-4)=0
(x^4+4)(x^2+2)(x^2-2)=0
to find the zeros, check each factor:
(x^4+4)=0
x^4= - 4
this has no solution.
(x^2+2)=0
x^2= - 2
this also has no solution, so the only answer is in
(x^2-2)=0
x^2=2
x=square root of 2 or x= -(square root of 2)
hence we have two solutions.
that's all

x^8 - 16 = 0
=> x^8 = 16
=> x^8 = (rt 2)^8
=> x = rt 2
or x = 1.414 to be approx

x^8-16=0
x^8 = 16
x=16^(1/8)
.:. x = 2^(1/2)
= 1.414.......

x^8 = 16

take square root of both sides to get

x^4 = 4

take square root of both sides again to get

x^2 = 2

take square root of both sides again to get

x = sqrt 2 = 1.414 or so

x^8-16=0
x^8 = 16
x^4 = 4
x^2 = 2
x = sqrt(2)
(= 1.4142...)

x^8-16=0
x^8 = 16

as x^mn = (x^m)^n

(x^2)^4= 2^4

thus

x^2=2

x= sqrt(2)
x=1.4142

HTH