Question

Find each root, if possible. $$\sqrt[3]{27}$$

   Find each root, if possible.
$$\sqrt[3]{27}$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 2nd Edition

Chapter 10, Problem 52

Instant Answer

Solved by Expert Ernest Castorena

06/26/2020

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Step 1: We are given the cube root of 27, which is denoted as $\sqrt[3]{27}$. Show more…

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Find each root, if possible. $$\sqrt[3]{27}$$

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Key Concepts

Cube Roots

Cube roots are the solutions to an equation of the form x³ = a. In the real number system, every number has exactly one real cube root. However, when working in the complex domain, a nonzero number has three distinct cube roots, which include one real and two complex values. This concept is part of solving polynomial equations and understanding inverse operations of cubing.

Principal Root

The principal root is the commonly accepted primary solution among multiple possible roots. For cube roots of a positive real number, the principal cube root is the positive real number that satisfies the equation x³ = a. This concept is used to provide a single, consistent answer in many contexts, especially when only real solutions are required.

Complex Roots

Complex roots arise from extending the idea of roots beyond the real number system. According to the fundamental theorem of algebra, a polynomial of degree n has n roots in the complex plane. For cube roots, even though one of the solutions may be real, the other two are complex and can be found using polar forms and De Moivre’s theorem. This approach ensures all possible solutions are accounted for in the complex number system.

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