Problem 30 - Contemporary Precalculus

Question

Find the nth roots in polar form. $$-16 \sqrt{2}-16 \sqrt{2} i ; \quad n=5$$

Step-by-Step Solution

Answer

Answer: The 5th roots of the given complex number in polar form are: 1. $$2 \operatorname{cis}\left(\frac{\pi}{20}\right)$$ 2. $$2 \operatorname{cis}\left(\frac{9\pi}{20}\right)$$ 3. $$2 \operatorname{cis}\left(\frac{17\pi}{20}\right)$$ 4. $$2 \operatorname{cis}\left(\frac{25\pi}{20}\right)$$ 5. $$2 \operatorname{cis}\left(\frac{33\pi}{20}\right)$$

Step 1: Convert the complex number into polar form

Given complex number: $$-16 \sqrt{2} - 16 \sqrt{2} i$$ To convert it into polar form, we need to find its magnitude and argument. Magnitude: $$r = \sqrt{(-16 \sqrt{2})^2 + (-16 \sqrt{2})^2} = \sqrt{512 + 512} = 32$$ Argument: $$\theta = \tan^{-1} \frac{-16 \sqrt{2}}{-16 \sqrt{2}} = \tan^{-1} (1) = \frac{\pi}{4}$$ Hence, the polar form of the complex number is: $$32 \operatorname{cis}\left(\frac{\pi}{4}\right)$$

Step 2: Apply De Moivre's theorem to find the nth roots

We need to find the 5th roots of the complex number in polar form, so we will use De Moivre's theorem, which states that to find the nth root of a complex number in polar form, we can simply divide its argument by n and distribute the root evenly: $$\sqrt[n]{A(\operatorname{cis}(\theta))}=\sqrt[n]{A}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)$$ Where k varies from 0 to n-1. In our case, n = 5, A = 32, and θ = π/4.

Step 3: Find the 5th roots of the complex number in polar form

Applying De Moivre's theorem as explained in step 2, we get: $$\sqrt[5]{32 \operatorname{cis}\left(\frac{\pi}{4}\right)}=\sqrt[5]{32}\operatorname{cis}\left(\frac{\frac{\pi}{4}+2k\pi}{5}\right)$$ Now we will find the roots for k = 0 to 4: For k = 0: $$2 \operatorname{cis}\left(\frac{\pi}{20}\right)$$ For k = 1: $$2 \operatorname{cis}\left(\frac{9\pi}{20}\right)$$ For k = 2: $$2 \operatorname{cis}\left(\frac{17\pi}{20}\right)$$ For k = 3: $$2 \operatorname{cis}\left(\frac{25\pi}{20}\right)$$ For k = 4: $$2 \operatorname{cis}\left(\frac{33\pi}{20}\right)$$ These are the 5th roots of the given complex number in polar form.

Key Concepts

Polar Form

Complex numbers can be expressed in two forms: rectangular and polar. Polar form is particularly useful in scenarios involving multiplication and finding powers or roots of complex numbers because it simplifies the process considerably.
Polar form represents a complex number using its magnitude (or modulus) and its angle (or argument). It is expressed as:\[ r \operatorname{cis}(\theta) \]Where:

• \( r \) is the magnitude, calculated as \( \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the real and imaginary parts respectively.
• \( \theta \) is the argument, which is the angle the line representing the complex number makes with the positive real axis, calculated as \( \tan^{-1}\left(\frac{y}{x}\right) \).

This form is compact and allows for easier manipulation of calculations involving multiplication, division, and finding powers or roots of complex numbers.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number analysis, making the computation of powers and roots of complex numbers both in polar form extremely straightforward.
It states that for any real number \( n \) and a complex number in polar form \( r \operatorname{cis}(\theta) \), the nth power is given by:\[(r \operatorname{cis}(\theta))^n = r^n \operatorname{cis}(n\theta)\]This theorem is particularly useful because it shows that we only need to raise the magnitude to the power and multiply the argument by the power. For finding roots, we rather divide the argument by the number of roots required and take the nth root of \( r \).
It simplifies calculations significantly compared to calculating powers using the rectangular form of complex numbers, thus making it a favored method in mathematics.

Nth Roots

Finding the nth root of a complex number, especially in polar form, is a commonly encountered problem in mathematics that De Moivre's Theorem aptly addresses. When you want to find the nth roots of a complex number, the theorem provides a straightforward method.
The process involves:

• Finding the nth root of the magnitude.
• Dividing the angle by \( n \) and adding \( \frac{2k\pi}{n} \) for each value of \( k \) from 0 to \( n-1 \) to obtain distinct roots.

The formula for finding the roots is:\[\sqrt[n]{r \operatorname{cis}(\theta)} = \sqrt[n]{r} \operatorname{cis}\left(\frac{\theta + 2k\pi}{n}\right)\]With these simple steps, the nth roots can be found effortlessly, providing solutions that incorporate both the length of the vector (magnitude) and its direction (argument) by making subtle adjustments at each step for k values. This method is efficient and elegant, offering clear geometrical insights into the structure of complex numbers.