i'm not sure about value of z^4+1/z^4

i only know that z^3=1

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If z^3 = 1, then z^3 - 1 = 0

We'll re-write the difference of cubes as:

z^3 - 1 = (z - 1)(z^2 + z + 1)

If z^3 - 1 = 0, then (z - 1)(z^2 + z + 1) = 0

We'll re-write the sum to be calculated as:

z^4 + 1/z^4 = z*z^3 + 1/z*z^3

But z^3 = 1 (1)

z^4 + 1/z^4 = z + 1/z

We'll multiply by z:

z^4 + 1/z^4 = (z^2 + 1)/z

But z^2 + z + 1 = 0 => z^2 + 1 = -z

z^4 + 1/z^4 = -z/z

z^4 + 1/z^4 = = -1

The requested value of the sum z^4 + 1/z^4 = -1**.**

If z^3 = 1 then let y = z^4 + 1/ z^4

Where y is the value you need to be sure of.

For z^3 = 1, z = 1^0.33 recurring.

1 raised to **any** power is equal to 1 therefore:

z = 1

Evaluating y = z^4 + 1/z^4

= 1^4 + 1/1^4

= 1 + 1/1

= 1+1 **= 2**

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