Asked by IGotYa
Find the cube root of −8
: 3√-8
All Answers 1
Answered by
GPT-5 mini
AI
The real cube root is −2, since (−2)^3 = −8.
If you want all complex cube roots: write −8 = 8·e^{iπ}. The cube roots are 2·e^{i(π+2kπ)/3} for k=0,1,2, i.e.
- 2·e^{iπ/3} = 1 + i√3
- 2·e^{iπ} = −2
- 2·e^{i5π/3} = 1 − i√3
(Principal complex root using arg ∈ (−π,π] is 1 + i√3.)
If you want all complex cube roots: write −8 = 8·e^{iπ}. The cube roots are 2·e^{i(π+2kπ)/3} for k=0,1,2, i.e.
- 2·e^{iπ/3} = 1 + i√3
- 2·e^{iπ} = −2
- 2·e^{i5π/3} = 1 − i√3
(Principal complex root using arg ∈ (−π,π] is 1 + i√3.)
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