Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

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

View step-by-step help

View step-by-step help

Evaluate rational exponents

Full solution

Q. Find all of the cube roots of

-1

and write the answers in rectangular (standard) form.

Identify Equation: Identify the equation for finding cube roots of a number. $\newline$ To find the cube roots of $-1$ , we solve the equation $x^3 = -1$ .
Solve Real Cube Root: Solve for the real cube root. $\newline$ The real cube root of $-1$ is $x = -1$ , because $(-1)^3 = -1$ .
Find Complex Roots: Find the complex cube roots using the polar form of complex numbers. $\newline$ Express $-1$ in polar form: $-1$ = $1$ *(cos(π) + i*sin(π)). $\newline$ Using De Moivre's Theorem, the nth roots are given by: $\newline$ $x_k = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right)$ $\newline$ for k = $0$ , $1$ , ..., n $-1$ , where n = $3$ here.
Calculate First Root: Calculate the first complex cube root (k= $1$ ). $\newline$ $x_1 = 1^{1/3} \left( \cos\left(\frac{\pi + 2\pi}{3}\right) + i\sin\left(\frac{\pi + 2\pi}{3}\right) \right)$ $\newline$ $x_1 = \cos\left(\frac{3\pi}{3}\right) + i\sin\left(\frac{3\pi}{3}\right)$ $\newline$ $x_1 = \cos(\pi) + i\sin(\pi)$ $\newline$ $x_1 = -1 + 0i$

More problems from Evaluate rational exponents

Question

\newline

81^{\frac{1}{2}}

\newline

Posted 2 months ago

Question

\newline

(-32)^{1/5}

\newline

Posted 2 months ago

Question

Simplify. Assume all variables are positive.

\newline

\frac{w^{\frac{4}{3}}}{w^{\frac{7}{3}}}

\newline

Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

Posted 2 months ago

Question

Simplify. Assume all variables are positive.

\newline

(2r^{\frac{1}{3}})^8

\newline

\newline

Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

Posted 1 month ago

Question

Simplify. Assume all variables are positive.

\newline

v^{\frac{7}{4}} \cdot v^{\frac{9}{4}}

\newline

\newline

Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

\newline

Posted 2 months ago

Question

\newline

1^{\frac{1}{4}} \cdot 1^{\frac{1}{4}}

\newline

Posted 1 month ago

Question

\newline

32^{(-1/5)}

Posted 1 month ago

Question

Simplify. Assume all variables are positive.

\newline

\frac{w^{\frac{3}{2}}}{w^{\frac{5}{2}}}

\newline

\newline

Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

\newline

Posted 2 months ago

Question

Simplify. Assume all variables are positive.

\newline

(27x)^{\frac{2}{3}}

\newline

\newline

Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

Posted 2 months ago

Question

Simplify. Assume all variables are positive.

\newline

\frac{x^{\frac{1}{4}}}{x^{\frac{11}{4}}}

\newline

\newline

Write your answer in the form

A

\frac{A}{B}

A

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

\newline

Posted 1 month ago