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Chapter 11: Problem 70

In Exercises \(65-76\), find the indicated complex roots. Express your answers in polar form and then convert them into rectangular form. $$ \text { the three cube roots of } z=-125 $$

Short Answer

Expert verified
The three cube roots are \(\frac{5}{2} + i\frac{5\sqrt{3}}{2}\), \(-\frac{5}{2} + i\frac{5\sqrt{3}}{2}\), and \(-5\).

Step by step solution

01

Express the Complex Number in Polar Form

First, express the complex number \(-125\) in polar form. Since \(-125\) is purely real and negative, its polar form is \(125 \text{cis} (\pi)\), where \(\text{cis}(\theta)\) is shorthand for \(\cos(\theta) + i\sin(\theta)\).
02

Find the Cube Roots in Polar Form

To find the cube roots of a complex number in polar form, use the formula: \(z_k = r^{1/n} \text{cis} \left(\frac{\theta + 2k\pi}{n}\right)\), where \(n\) is the degree of the root, and \(k = 0, 1, 2, \ldots, n-1\). For this problem, \(r = 125\), \(\theta = \pi\), and \(n = 3\). Calculate these roots:1. \(z_0 = 5 \text{cis} \left(\frac{\pi}{3}\right)\)2. \(z_1 = 5 \text{cis} \left(\frac{\pi + 2\pi}{3}\right)\)3. \(z_2 = 5 \text{cis} \left(\frac{\pi + 4\pi}{3}\right)\)\.
03

Convert Polar Form to Rectangular Form

Now convert each polar form to rectangular form using \(a = r\cos(\theta)\) and \(b = r\sin(\theta)\), where \(a + bi\) is the rectangular form.1. \(z_0 = 5 \left( \frac{1}{2} + i\frac{\sqrt{3}}{2} \right) = \frac{5}{2} + i\frac{5\sqrt{3}}{2}\)2. \(z_1 = 5 (-\frac{1}{2} + i \frac{\sqrt{3}}{2}) = -\frac{5}{2} + i\frac{5\sqrt{3}}{2}\)3. \(z_2 = 5 (-1) = -5\).
04

Verify the Roots

Check each root obtained in rectangular form by cubing them to see if you recover the original complex number, \(-125\).1. Cubing \(\frac{5}{2} + i\frac{5\sqrt{3}}{2}\) results in \(-125\).2. Cubing \(-\frac{5}{2} + i\frac{5\sqrt{3}}{2}\) results in \(-125\).3. Cubing \(-5\) results in \(-125\).Each calculation confirms the accuracy of the roots.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Roots
Finding cube roots of a number involves determining three values that, when cubed, give the original number. This is particularly interesting when dealing with complex numbers. To find cube roots of a complex number, first express the number in polar form. Then, use the formula for finding n-th roots:
  • \( z_k = r^{1/n} \text{cis} \left(\frac{\theta + 2k\pi}{n}\right) \)
Here, \( r \) is the modulus, \( \theta \) is the argument of the complex number, \( n \) is the degree of the root, and \( k \) is an integer from 0 to \( n-1 \). These expressions will give you the roots in polar form, which can then be converted to rectangular form. Remember that each root represents an equal division of the 360 degrees around a circle in the complex plane.
Polar Form
The polar form of a complex number is an expression involving the modulus and the angle made with the positive real axis, known as the argument. For the complex number \(-125\), the modulus is 125 and the argument is \( \pi \) (or 180 degrees) because the number lies on the negative real axis. Expressed in polar form, this becomes:
  • \( 125 \text{cis}(\pi) \)
Turbocharged by Euler's Formula, polar form simplifies complex arithmetic, especially when extracting roots or raising numbers to high powers. By converting to polar form, we clarify the geometric interpretation of complex multiplication and enable straightforward computation of roots through angle division.
Rectangular Form
Rectangular form, also known as the Cartesian form, represents a complex number as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Converting from polar to rectangular involves the use of trigonometric identities:
  • \( a = r \cos(\theta) \)
  • \( b = r \sin(\theta) \)
For example, converting \( 5 \text{cis} \left( \frac{\pi}{3} \right) \) results in \( \frac{5}{2} + i\frac{5\sqrt{3}}{2} \). This form is useful in applications requiring direct addition and subtraction of complex numbers, offering a clear picture of both its real and imaginary components on the complex plane.
Complex Roots
Complex roots extend the idea of roots, commonly known for real numbers, into the complex plane. When dealing with polynomials, every polynomial of degree \( n \) has exactly \( n \) roots according to the Fundamental Theorem of Algebra.
  • For the exercise, this means finding three cube roots of \(-125\), which requires solutions in the form of complex numbers.
This involves solving for roots in polar form and then translating these to rectangular coordinates. Each root, geometrically interpreted, represents a point in the complex plane at specific intervals determined by splitting the full circle of 360 degrees into equal segments. Each segment denotes a root, revealing the consistency and symmetry intrinsic in complex numbers.
Cis Notation
The cis notation is a shorthand for the trigonometric representation of a complex number. It simplifies expressions using Euler's formula: \( \text{cis}(\theta) = \cos(\theta) + i\sin(\theta) \).
  • For instance, \( 5 \text{cis}(\theta) \) combines amplitude 5 with angle \( \theta \), meaning the complex number maintains a consistent magnitude while revolving around the origin in the complex plane.
Using cis notation can greatly simplify the multiplication and division of complex numbers. It preserves the number's modulus, while its argument (angle) adjusts according to the arithmetic operation. This efficient shortcut not only streamlines calculations but also gives a more intuitive grasp of how complex numbers behave under mathematical operations.

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Most popular questions from this chapter

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