Which expression represents a 5th root of –i? cos(7pie/4)+sin(7pie/4)
cos(7pie/6)+sin(7pie/6)
cos(7pie/8)+sin(7pie/8)
cos(7pie/10)+sin(7pie/10)

The 5th root of -i is (d)

By DeMoivre's formula, we have:

The 5th root of -i means that: n = 5.

So, we have:

Set to any expression such that:

So, we have:

Expand

Take 5th roots of both sides

This means that:

is the 5th root of -i

Hence, option (d) is correct

Read more about complex numbers at:

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Use DeMoivre's formula:

(cos(θ) + i sin(θ))ⁿ = cos(n θ) + i sin(n θ)

If a given number is a 5th root of -i, raising it to the 5th power should return -i.

Since -i = cos(-π/2) + i sin(-π/2), you should check any number with an argument θ such that 5θ = -π/2. This happens only for the last choice:

(cos(7π/10) + i sin(7π/10))⁵ = cos(35π/10) + i sin(35π/10)

and 35π/10 is equivalent to -π/2 (modulo 2π).

the least value out of the data shown is 5, maybe if you post a picture with it i could a little more.

step-by-step explanation: