Problem 111

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$2(\sqrt{3}-i)^{5}$$

Step-by-Step Solution

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Answer

The fifth power of the complex number $2(\sqrt{3}-i)$ is $32\sqrt{3} - 32i$.

1Step 1: Convert to trigonometric form

A complex number can be written in the form $ r(\cos\theta + i\sin\theta)$, where $ r$ is the modulus of the complex number and $ \theta$ is the argument. The modulus, $ r$, is calculated as $\sqrt{a^2 + b^2}$, where a and b are the real and imaginary parts of the complex number respectively. The argument, $ \theta$, can be found using the formula $ \tan^{-1} (b/a)$. Thus, for the complex number $\sqrt{3} - i$, we have $ r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2$ and $ \theta = \tan^{-1}(-1/\sqrt{3}) = -\pi/6$. So the complex number in trigonometric form is $ 2(\cos(-\pi/6) + i\sin(-\pi/6))$.

2Step 2: Apply DeMoivre's Theorem

According to DeMoivre's Theorem, to find the 5th power of this complex number, simply multiply the angle by 5 and raise $ r$ to the fifth power: $ 2^5(\cos 5(-\pi/6) + i\sin 5(-\pi/6)) = 32(\cos(-5\pi/6) + i\sin(-5\pi/6))$.

3Step 3: Convert back to standard form

To convert back to standard form, evaluate the trigonometric functions: $ 32(\cos(-5\pi/6) + i\sin(-5\pi/6)) = 32((\sqrt{3}/2) - i/2) = 16\sqrt{3} - 16i$.

4Step 4: Multiply by given constant

Finally, multiply this result by the original given constant 2, getting $2(16\sqrt{3} - 16i) = 32\sqrt{3} - 32i$.

Key Concepts

Complex NumbersTrigonometric FormStandard FormModulus and Argument

Complex Numbers

Complex numbers are a fundamental concept in mathematics, which extend the idea of one-dimensional real numbers to two dimensions. They can be expressed in the form of $a + bi$, where $a$ is the real part and $b$ is the imaginary part multiplied by the imaginary unit $i$. Here, $i$ is defined such that $i^2 = -1$, allowing for the representation of all numbers.

Real Part: The component $a$ of the complex number that lies on the horizontal axis (real axis).
Imaginary Part: The component $b$, which includes $i$, plotted on the vertical axis (imaginary axis).
Imaginary Unit: $i$, where $i^2 = -1$.

This dual nature allows complex numbers to handle operations that have no solutions in the real domain, such as the square roots of negative numbers.

Trigonometric Form

Trigonometric form is a way to represent complex numbers using polar coordinates. This is incredibly useful for multiplying, dividing, and raising complex numbers to powers, due to the relationship with angles. A complex number $z = a + bi$ can be expressed as $r(\cos \theta + i\sin \theta)$, where $r$ is the modulus and $\theta$ is the argument.

Modulus $r$: Represents the distance of the complex number from the origin in the complex plane.
Argument $\theta$: Represents the angle the complex number makes with the positive real axis.

Switching to trigonometric form allows us to utilize DeMoivre's Theorem effectively, simplifying many operations on complex numbers.

Standard Form

The standard form of a complex number is the Cartesian form given as $a + bi$. This form is straightforward, allowing easy addition, subtraction, and interpretation. It's easily related to graphing, as it directly reflects operations in a standard xy-plane.

Addition and Subtraction: Performed component-wise, combining like terms.
Graphical Representation: Easier to visualize as points or vectors on the complex plane.
Simplification: Helpful for presenting the final results, like in the given problem where $2(16\sqrt{3} - 16i)$ becomes $32\sqrt{3} - 32i$.

This form becomes particularly useful after calculations in trigonometric form, as it gives interpretable results.

Modulus and Argument

The modulus and argument are crucial components when dealing with complex numbers, especially in their trigonometric form. They help in rewriting the complex number in a way that is optimal for exponentiation and root calculations.