Question

Find each root, if possible. $$\sqrt[3]{-8}$$

   Find each root, if possible.
$$\sqrt[3]{-8}$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 2nd Edition

Chapter 10, Problem 54

Instant Answer

Solved by Expert Ernest Castorena

06/26/2020

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Step 1: We are given the cube root of -8 and we need to find the number which when cubed gives -8. Show more…

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Find each root, if possible. $$\sqrt[3]{-8}$$

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Key Concepts

Polar Representation of Complex Numbers

Expressing complex numbers in polar form, as a magnitude and an angle, simplifies the process of finding roots. This representation is beneficial because it transforms multiplication and exponentiation into operations involving the magnitude and angle, making it easier to distribute roots evenly along a circle.

De Moivre's Theorem

De Moivre's Theorem relates complex numbers in polar form to exponentiation and root extraction. It states that raising a complex number to a power involves raising its modulus to that power and multiplying its argument by the power. When extracting roots, the theorem is used to divide the angle by the root's degree, generating all possible roots by including multiples of the full rotation divided by the degree of the root.

Cube Roots

A cube root of a number is a value that, when cubed, results in the original number. For any given number, there exists one real cube root and, if complex numbers are considered, two additional complex cube roots. This concept is central to understanding how roots are extracted for polynomial equations of degree three.

Complex Roots and Their Properties

In the context of complex numbers, every non-zero number has three distinct cube roots. These roots are evenly spaced around the unit circle in the complex plane. Recognizing that multiple values exist when extracting odd roots is important in many areas of mathematics, especially when solving polynomial equations.

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