Question

Find each root. See Example 1. $$\sqrt[5]{32}$$

   Find each root. See Example 1.
$$\sqrt[5]{32}$$

Algebra for College Students

Algebra for College Students

Mark Dugopolski 5th Edition

Chapter 7, Problem 17

Instant Answer

Solved by Expert Joshua Eastwood

04/14/2020

Step 1

This means we need to find a number that, when raised to the power of 5, equals 32. Show more…

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Find each root. See Example 1. $$\sqrt[5]{32}$$

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Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra asserts that every non-zero polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities. This concept guarantees that even if a number like 32 appears as a straightforward positive real number, the equation x^5 = 32 has five roots, including complex solutions.

Polar Form Representation

Polar form representation expresses complex numbers in terms of their magnitude and angle relative to the positive real axis. This form is particularly useful when performing operations such as taking powers and roots, as it simplifies multiplication, division, and exponentiation into operations on the magnitude and angle separately.

nth Roots

nth roots involve finding all solutions to the equation x^n = a, where n is a positive integer and a can be any number, real or complex. When a is expressed in polar form, the nth roots can be derived by taking the nth root of the magnitude and evenly distributing the angle by adding multiples of 2?, ensuring all distinct solutions are captured.

De Moivre's Theorem

De Moivre's Theorem is a fundamental principle for working with complex numbers in polar form. It states that for any real number ? and integer n, (cos? + i sin?)^n equals cos(n?) + i sin(n?). This theorem facilitates finding nth roots of complex numbers by allowing the conversion between exponential and trigonometric representations.

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