Question

\(4 \sqrt{(-8+8 \sqrt{3} i)}=\) (a) \(\pm(1+\sqrt{3 i})\) (b) \(\pm(2+2 \sqrt{3 i})\) (c) \(\pm(\sqrt{3}+\mathrm{i})\) (d) \(\pm(2-2 \sqrt{3} \mathrm{i})\)

Short answer

The short answer is: \( \pm(2-2\sqrt{3}i) \) (option d).

Step-by-step solution

Write the complex number in polar form.

First, we need to rewrite the given complex number in polar form. To do this, we can first express the number inside the square root in rectangular form using the real part and the imaginary part. So we have: \(-8+8\sqrt{3}i\) Now, we need to find the magnitude (r) and argument (θ) of the complex number. We can use the following formulas to find r and θ: \(r = \sqrt{a^2+b^2}\) \(\theta = \arctan{\frac{b}{a}}\) where a and b are the real and imaginary parts of the complex number. In our case, we have a = -8 and b = 8√3. So we have: \(r = \sqrt{(-8)^2+(8\sqrt{3})^2} = \sqrt{64+192} = \sqrt{256} = 16\) \(\theta = \arctan{\frac{8\sqrt{3}}{-8}} = \arctan{-\sqrt{3}} = \frac{4\pi}{3}+2k\pi \rm{, k\in Z}\) since arguments are only unique up to integer multiples of \(2\pi\) Hence, the polar form of the given complex number is: \(16\left(\cos{\frac{4\pi}{3}}+i\sin{\frac{4\pi}{3}}\right)\). Now, we can rewrite the given expression as: \(4\sqrt{16\left(\cos{\frac{4\pi}{3}}+i\sin{\frac{4\pi}{3}}\right)}\)

Apply the De Moivre's theorem to simplify the radical expression.

Applying De Moivre's theorem, we obtain: \(4\left(\cos{\frac{\theta}{2}}+i\sin{\frac{\theta}{2}}\right)\), where \(\theta =\frac{4\pi}{3}+2k\pi \rm{, k\in Z}\) So, substituting for \(\theta\): \(4\left(\cos{\frac{2\pi}{3}+k\pi}+i\sin{\frac{2\pi}{3}+k\pi}\right)\).

Convert the result back to rectangular form.

Now, the expression is in trigonometric form. Converting the polar form to rectangular form, we use the standard equations: \(a=r\cos\theta\) \(b=r\sin\theta\) So the rectangular form is: \(4\left(2\cos\left(\frac{2\pi}{3}+k\pi\right)+2i\sin\left(\frac{2\pi}{3}+k\pi\right) \right)\) Considering integer values of k, for even numbers (k = 0, 2, -2, ...), the expression becomes: \(4\left(2\cos\left(\frac{2\pi}{3}\right)+2i\sin\left(\frac{2\pi}{3}\right) \right)=4\left(-1 + \sqrt{3} i\right) = -4 + 4\sqrt{3}i\) And for odd values of k (k = 1, -1, 3, ...), the expression becomes: \(4\left(2\cos\left(\frac{5\pi}{3}\right)+2i\sin\left(\frac{5\pi}{3}\right) \right)=4\left(-1 - \sqrt{3} i\right) = -4 - 4\sqrt{3}i\) From this, we can summarize that the result is \(\pm(2-2\sqrt{3}i)\).

Match the answer with one of the given options.

Our final answer is \(\pm(2-2\sqrt{3}i)\). Comparing this with the options, we can see that it matches option (d). Hence, the correct choice is (d).

Key concepts

Polar Form

Complex numbers can be represented in two primary ways: rectangular form and polar form. The polar form leverages the concept of expressing a complex number as a vector. This form provides a way to represent the number through its magnitude and direction.

In polar form, a complex number is expressed as \( r(\cos{\theta} + i\sin{\theta}) \), where:

• \( r \) is the magnitude, calculated as \( \sqrt{a^2 + b^2} \) with \( a \) and \( b \) being the real and imaginary parts, respectively.
• \( \theta \) is the argument, found using \( \arctan(\frac{b}{a}) \).

Using polar form provides significant advantages in multiplication and division of complex numbers, as well as in computing powers and roots, because operations are simplified when the numbers are expressed in terms of angles and radii.

In the given exercise, re-formulating the complex number \(-8 + 8\sqrt{3}i\) using its magnitude of 16 and an angle of \( \frac{4\pi}{3} \) allows easier subsequent manipulations.

De Moivre's Theorem

De Moivre's theorem is a powerful tool for calculating powers and roots of complex numbers when they are in polar form. It states that for any complex number \( z = r(\cos{\theta} + i\sin{\theta}) \) and any integer \( n \), \[(z)^n = r^n (\cos{n\theta} + i\sin{n\theta})\]

This theorem simplifies the process of exponentiating a complex number by turning it into a simple manipulation of the angle \( \theta \) and the magnitude \( r \).

In the exercise's step-by-step solution, applying De Moivre's theorem allows us to handle the square root of a complex number efficiently, by dividing the angle \( \theta \) by 2 (as we are seeking the square root, which corresponds to a power of \( \frac{1}{2} \)). This results in the new angle being \( \frac{2\pi}{3} \). The original magnitude 16 is neatly handled by recognizing \( 4\sqrt{16} \) simplifies into manageable expressions leveraging this theorem.

Rectangular Form

Rectangular or Cartesian form of a complex number is simply \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part. This format is straightforward, representing the complex number as a point in the two-dimensional plane.

In the context of converting from polar to rectangular form, we utilize the relationships:

• \( a = r\cos{\theta} \)
• \( b = r\sin{\theta} \)

This conversion provides practical value, such as finding the precise numeric values of the components that makeup complex numbers, suitable for computations, or when integration with other algebraic expressions is required.

In the exercise, once De Moivre's theorem has been applied, converting back to rectangular form confirms the numeric expression \(-4 \pm 4\sqrt{3}i\), leading us to match with the right option \( \pm(2 - 2\sqrt{3}i) \). It becomes evident that careful navigation between polar and rectangular forms is instrumental for solving complex equation exercises efficiently.