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Find all of the cube roots of -1 and write the answers in rectangular (standard) for

Find all of the cube roots of $-1$ and write the answers in rectangular (standard) for

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Evaluate rational exponents

Full solution

Q. Find all of the cube roots of

-1

and write the answers in rectangular (standard) for

Identify Equation: Identify the equation for finding cube roots of a number. $\newline$ Equation: $x^3 = -1$
Solve Real Root: Solve for one real root using the fact that if $x^3 = -1$ , then $x = -1$ . $\newline$ Calculation: $(-1)^3 = -1$
Identify Complex Roots: Identify the complex roots using the polar form of complex numbers. $\newline$ For $x^3 = -1$ , the polar form is $re^{i\theta}$ where $r = 1$ and $\theta = \frac{2\pi k}{3}$ for $k = 1, 2$ .
Calculate First Root: Calculate the first complex root for $k = 1$ . $\newline$ $\theta = \frac{2\pi}{3}$ $\newline$ $x = e^{i\frac{2\pi}{3}} = \cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$
Calculate Second Root: Calculate the second complex root for $k = 2$ . $\newline$ $\theta = \frac{4\pi}{3}$ $\newline$ $x = e^{i\frac{4\pi}{3}} = \cos(\frac{4\pi}{3}) + i\sin(\frac{4\pi}{3}) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$

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