Problem 27 - Advanced Engineering Mathematics

Question

In Problems 27-32, use (10) to compute all roots. Sketch these roots on an appropriate circle centered at the origin. $$ (8)^{1 / 3} $$

Step-by-Step Solution

Answer

The cube roots of 8 are 2, \(-1 + i\sqrt{3}\), and \(-1 - i\sqrt{3}\).

Step 1: Express the Problem in Exponential Form

The given problem is to find the cube roots of 8. We start by expressing 8 in an exponential form: \[ 8 = 2^3 \] Thus, \[ (8)^{1/3} = (2^3)^{1/3} = 2 \] This gives us the principal root. However, since we are dealing with cube roots, there are two additional complex roots to consider.

Step 2: Apply De Moivre's Theorem

To find all cube roots, we use the formula for the roots of a complex number, \[ z^{1/n} = |z|^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \] For the problem at hand, express 8 in polar form as 8:\[ 8 = 8(\cos 0 + i\sin 0) \] Now apply the formula for cube roots (\(n=3\)):

Step 3: Calculate All Cube Roots

Calculate the three roots:1. For \(k=0\), \[ 2\left( \cos\left(\frac{0}{3}\right) + i\sin\left(\frac{0}{3}\right) \right) = 2 \]2. For \(k=1\), \[ 2\left( \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) \right) = 2\left( -\frac{1}{2} + i\frac{\sqrt{3}}{2} \right) = -1 + i\sqrt{3} \]3. For \(k=2\), \[ 2\left( \cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right) \right) = 2\left( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \right) = -1 - i\sqrt{3} \]

Step 4: Sketch the Roots on the Complex Plane

With the roots obtained: 2, \(-1 + i\sqrt{3}\), and \(-1 - i\sqrt{3}\), plot these on the complex plane. The roots all lie on a circle of radius 2 centered at the origin (0, 0). Each root corresponds to a point on the circle:- Root 1 at 2 on the real axis.- Root 2 at \(-1 + i\sqrt{3}\) in the upper half plane.- Root 3 at \(-1 - i\sqrt{3}\) in the lower half plane.

Key Concepts

Cube Roots

Cube roots are values that, when raised to the power of 3, give the original number. For example, the cube roots of 8 are numbers that when multiplied by themselves three times result in 8. This concept involves real numbers as well as complex numbers.
One important thing to remember is that every real number can have three cube roots. Of these, one is real, and the others are complex.

• The principal root: This is the simplest root, often the real number root. For 8, its principal root is 2.
• Complex roots: These are calculated using complex numbers, often resulting in two additional roots.

Calculating all cube roots involves understanding both the principal root and utilizing additional mathematical tools like De Moivre's theorem to find complex roots.

De Moivre's Theorem

De Moivre's Theorem is key when dealing with roots of complex numbers. It provides a formula that links complex numbers with trigonometry. This theorem helps us find all nth roots, including cube roots, for complex numbers.

The theorem states that for any complex number expressed in polar form,

• The roots are given by: \[ (z^{1/n}) = |z|^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \]
• Here, \(k\) is an integer that takes values from 0 up to \(n-1\), allowing us to find all possible roots.

In our example with 8, the number is seen as a complex number 8(cos 0 + i*sin 0), and the theorem helps find the cube roots by rotating the angle for each root.

Polar Form

Polar form is a way of representing complex numbers. Unlike the regular form which uses real and imaginary parts, polar form uses magnitude (distance from the origin) and angle (direction from the positive real axis).

• Magnitude: Represented as \(|z|\), it is the distance from the origin to the point corresponding to the complex number on the complex plane.
• Angle: Also known as the argument \(\theta\), it is the angle formed with the positive real axis.

To convert a complex number to polar form, use:\[ z = r(\cos \theta + i \sin \theta)\]In our example, 8 is firmly on the real axis, so its polar form is expressed with magnitude 8 and angle 0. This form is crucial for applying De Moivre's Theorem and calculating roots.

Complex Plane

The complex plane is used to visualize complex numbers graphically. Here, every complex number corresponds to a point on this plane.

• Horizontal Axis: Represents the real part of the complex number.
• Vertical Axis: Represents the imaginary part.

The complex plane is essential when sketching the roots of complex numbers. Plotting our roots for 8 shows:- Two lies on the real axis at position 2.- The second root \(-1 + i\sqrt{3}\) places it on the upper half-plane.- The third root \(-1 - i\sqrt{3}\) is positioned on the lower half-plane.These roots land on a circle centered at the origin with a radius equal to the magnitude of the principal root, which illustrates how complex numbers and their roots relate spatially.