Answer :
Answer:
{8, -4 +i·4√3, -4 -i·4√3}
Step-by-step explanation:
Using Euler's formula, we can find the roots from ...
512 = 8^3 = 8^3·e^(2nπi)
Then the cube root is ...
root = 512^(1/3) = 8e^(2nπi/3)
The values will be different for n = 0, 1, 2; then they repeat. So the three cube roots of 512 are ...
- 8
- 8e^(2/3πi)
- 8e^(4/3πi)
The latter two can be written as the complex numbers ...
8e^(2/3πi) = 8(cos(2/3π) +i·sin(2/3π)) = -4 +i·4√3
and
8e^(4/3πi) = 8(cos(4/3π) +i·sin(4/3π)) = -4 -i·4√3
Then the three cube roots of 512 are ...
{8, -4 +i·4√3, -4 -i·4√3}
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These are evenly spaced around the circle with radius 8 in the complex plane.