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Find all of the cube roots of -1 and write the answers in rectangular (standard) form.
Answer Attempt 2 out of 2

Find all of the cube roots of $-1$ and write the answers in rectangular (standard) form. $\newline$ Answer Attempt $2$ out of $2$

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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) form.

\newline

Answer Attempt

2

out of

2

Identify Equation: Identify the equation for finding cube roots of a number. $\newline$ To find the cube roots of $-1$ , we solve the equation $x^3 = -1$ .
Convert to Polar Form: Convert the equation into polar form to facilitate finding roots. $\newline$ The number $-1$ can be represented in polar form as $1*(\cos(\pi) + i*\sin(\pi))$ .
Apply De Moivre's Theorem: Apply De Moivre's Theorem to find the cube roots. $\newline$ Using De Moivre's Theorem, the nth roots (n= $3$ here) of a complex number r(cos(θ) + i*sin(θ)) are given by: $\newline$ $r^{1/n} \left( \cos\left(\frac{θ + 2kπ}{n}\right) + i \sin\left(\frac{θ + 2kπ}{n}\right) \right)$ $\newline$ for k = $0$ , $1$ , ..., n $-1$ .
Calculate First Root: Calculate the first root (k= $0$ ). $\newline$ Plug k= $0$ into the formula: $\newline$ $1^{1/3} \left( \cos\left(\frac{π + 2*0*π}{3}\right) + i \sin\left(\frac{π + 2*0*π}{3}\right) \right)$ $\newline$ = cos(π/ $3$ ) + i*sin(π/ $3$ ) $\newline$ = $1$ / $2$ + i*√ $3$ / $2$
Calculate Second Root: Calculate the second root (k= $1$ ). $\newline$ Plug k= $1$ into the formula: $\newline$ $1^{1/3} \left( \cos\left(\frac{π + 2*1*π}{3}\right) + i \sin\left(\frac{π + 2*1*π}{3}\right) \right)$ $\newline$ = cos(π) + i*sin(π) $\newline$ = $-1$ + i* $0$
Calculate Third Root: Calculate the third root (k= $2$ ). $\newline$ Plug k= $2$ into the formula: $\newline$ $1^{1/3} \left( \cos\left(\frac{π + 2*2*π}{3}\right) + i \sin\left(\frac{π + 2*2*π}{3}\right) \right)$ $\newline$ = cos( $5$ π/ $3$ ) + i*sin( $5$ π/ $3$ ) $\newline$ = $1$ / $2$ - i*√ $3$ / $2$

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