Problem 80 - Precalculus with Limits

Question

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((\sqrt{5}\ -\ 4i)^3\)

Step-by-Step Solution

Answer

The result of \((\sqrt{5} -4i)^3\) is \((\sqrt{21})^3\cos(3\arctan(\frac{-4}{\sqrt{5}})) + (\sqrt{21})^3\sin(3\arctan(\frac{-4}{\sqrt{5}}))i\)

Step 1: Convert to Polar Form

In a complex number \(a+bi\), \(a\) can be found as \(r\cos \theta\), and \(b\) can be found as \(r\sin \theta\). Hence, first there is a need to compute \(r = \sqrt{a^2 + b^2}\) and \(\theta = \arctan(\frac{b}{a})\). For \(\sqrt{5} -4i\), \(r = \sqrt{(\sqrt{5})^2 + (-4)^2}= \sqrt{5+16} = \sqrt{21}\) and \(\theta = \arctan(\frac{-4}{\sqrt{5}})\)

Step 2: Apply DeMoivre's Theorem

Use the formula \((r\cos \theta + r\sin \theta i)^n = r^n (\cos (n \theta) + i\sin (n\theta))\), where n = 3, r = \(\sqrt{21}\) and \(\theta = \arctan(\frac{-4}{\sqrt{5}})\). This will yield \((\sqrt{21})^3(\cos(3\arctan(\frac{-4}{\sqrt{5}})) + i\sin(3\arctan(\frac{-4}{\sqrt{5}})))\)

Step 3: Convert back to Standard Form

Now, convert the polar form back to \(a+bi\) form, where \(a=r\cos \theta\) and \(b=r\sin \theta\). Thus we get the final answer of \((\sqrt{21})^3\cos(3\arctan(\frac{-4}{\sqrt{5}})) + (\sqrt{21})^3\sin(3\arctan(\frac{-4}{\sqrt{5}}))i\)

Key Concepts

Understanding Complex Numbers

Complex numbers are a fundamental concept in mathematics that extend the real numbers with an additional dimension, allowing the solution of equations that have no real solutions. At the heart of a complex number is the imaginary unit \( i \), which is defined by \( i^2 = -1 \). A complex number is expressed as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

For example, in the complex number \( 3 + 4i \), \( 3 \) is the real part and \( 4i \) is the imaginary part. This allows for operations and functions that are not possible with just real numbers, such as taking the square root of a negative number. Complex numbers are crucial in many fields, notably in engineering and physics, where they are used to describe electrical circuits, waveforms, and quantum states.

Polar Form of Complex Numbers

The polar form is another way to represent complex numbers and is particularly useful when dealing with multiplication, division, and powers of complex numbers. A complex number in polar form is written as \( r(\text{cos} \theta + i\text{sin} \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument) with the positive direction of the x-axis.

To find \( r \), we use the formula \( r = \sqrt{a^2 + b^2} \), which comes from the Pythagorean theorem. The angle \( \theta \) is found by \( \theta = \arctan(\frac{b}{a}) \), or by using a combination of trigonometric functions if the ratio lies outside the range of arctan. The polar form provides a clear geometric interpretation of multiplication and division, as these operations correspond to multiplying magnitudes and adding angles, or dividing magnitudes and subtracting angles, respectively.

Powers of Complex Numbers using DeMoivre's Theorem

DeMoivre's Theorem is a powerful and elegant tool in mathematics used to calculate powers of complex numbers. It states that the nth power of a complex number in polar form, \( r(\cos \theta + i\sin \theta) \), is given by \( r^n (\text{cos}(n\theta) + i\text{sin}(n\theta)) \). This theorem simplifies the computation of complex powers, which would be otherwise more tedious through multiplication in standard form.

To apply DeMoivre's Theorem, we first convert the complex number into polar form. Then we raise the magnitude \( r \) to the power of \( n \) and multiply the angle \( \theta \) by \( n \). Finally, we convert the result back to standard form if needed, utilizing the property that \( \text{cos}(\theta) = \frac{a}{r} \) and \( \text{sin}(\theta) = \frac{b}{r} \) to find the real and imaginary parts respectively. Utilizing DeMoivre's Theorem is not only computationally efficient but also provides deep insight into the behavior of complex numbers raised to integer powers.