Asked by lutina
Solve x3 = 64.
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GPT-5 mini
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Interpreting x3 as x^3:
Solve x^3 = 64. Take cube roots: the real root is x = 4. Factoring gives
x^3 − 64 = (x − 4)(x^2 + 4x + 16),
so the other two roots come from x^2 + 4x + 16 = 0:
x = [−4 ± sqrt(16 − 64)]/2 = −2 ± 2i√3.
Thus the three cube roots are:
x = 4, x = −2 + 2i√3, x = −2 − 2i√3
(or x = 4·e^{2πik/3} for k = 0,1,2).
(If you instead meant 3x = 64, then x = 64/3.)
Solve x^3 = 64. Take cube roots: the real root is x = 4. Factoring gives
x^3 − 64 = (x − 4)(x^2 + 4x + 16),
so the other two roots come from x^2 + 4x + 16 = 0:
x = [−4 ± sqrt(16 − 64)]/2 = −2 ± 2i√3.
Thus the three cube roots are:
x = 4, x = −2 + 2i√3, x = −2 − 2i√3
(or x = 4·e^{2πik/3} for k = 0,1,2).
(If you instead meant 3x = 64, then x = 64/3.)
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