Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$(-1+i)^{7}$$

Step-by-Step Solution

Answer

The rectangular form of \((-1+i)^{7}\) is \(-8 - 8i\).

Step 1: Express in Polar Form

The complex number \(-1 + i\) can be written in polar form. First, calculate the modulus \(r\) using the formula \(r = \sqrt{a^2 + b^2}\), where \(a = -1\) and \(b = 1\). Thus, \(r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\).

Step 2: Calculate Argument

The argument \(\theta\) of the complex number \(-1 + i\) is the angle whose tangent is \(\tan\theta = \frac{b}{a} = \frac{1}{-1} = -1\). In the complex plane, the point is in the second quadrant, giving \(\theta = \frac{3\pi}{4}\).

Step 3: Apply De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), \((r (\cos \theta + i \sin \theta))^n = r^n (\cos (n\theta) + i \sin (n\theta))\). Here, \(r = \sqrt{2}\) and \(\theta = \frac{3\pi}{4}\), so:\[(-1 + i)^7 = (\sqrt{2})^7 \left(\cos \left(7 \times \frac{3\pi}{4}\right) + i \sin \left(7 \times \frac{3\pi}{4}\right)\right)\]

Step 4: Simplify Modulus and Argument

Calculate \((\sqrt{2})^7 = 2^{3.5} = 8\sqrt{2}\). Simplify the angle:\[7 \times \frac{3\pi}{4} = \frac{21\pi}{4} = \frac{5\pi}{4} + 2\pi\]Subtract \(2\pi\) to find an equivalent angle:\(\frac{5\pi}{4}\).

Step 5: Express Result in Rectangular Form

Now evaluate:\[8\sqrt{2}(\cos(\frac{5\pi}{4}) + i \sin(\frac{5\pi}{4}))\]Since \(\cos(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}\) and \(\sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}\), we have:\[8\sqrt{2} \left(-\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) = 8(-1 - i) = -8 - 8i\]

Step 6: Check and Finalize

Combine the results to finalize the answer. The rectangular form is: \(-8 - 8i\). This confirms the power is evaluated correctly based on the polar form and De Moivre's theorem.

Key Concepts

Polar Form

The polar form of a complex number is a way of representing the number mathematically based on its magnitude and angle from the positive real axis. It is often expressed as \[ r(\cos \theta + i \sin \theta) \] where:

• \( r \) is the modulus or magnitude, calculated using \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts, respectively.
• \( \theta \) is the argument or angle, typically found using \( \tan \theta = \frac{b}{a} \).

For example, the complex number \(-1 + i\) can be rewritten in polar form by calculating its modulus \( r = \sqrt{2} \) and its argument \( \theta = \frac{3\pi}{4} \).Understanding polar form is essential for performing operations like multiplication and division of complex numbers easily in their polar representations.

De Moivre's Theorem

De Moivre's Theorem is a critical concept when dealing with powers and roots of complex numbers in polar form. It states that for any complex number in polar form, \[ (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \]This theorem greatly simplifies the process of finding powers of complex numbers. Instead of multiplying the complex number by itself repeatedly, it allows us to raise the modulus to the power and multiply the angle by the exponent directly.
For instance, to calculate \((-1 + i)^7\),we first convert \(-1 + i\)to polar form, then apply De Moivre's Theorem,resulting in\((\sqrt{2})^7(\cos(\frac{21\pi}{4}) + i\sin(\frac{21\pi}{4}))\). This expression can then be further simplified using trigonometric identities and properties of angles, demonstrating the power of De Moivre’s Theorem.

Rectangular Form

Rectangular, or Cartesian form, represents complex numbers using their real and imaginary components. It is written as \( a + bi \),where:

• \( a \) is the real part.
• \( b \) is the imaginary part.

To transition from polar to rectangular form, one must evaluate the trigonometric functions to find precise values of real and imaginary components.
For the problem \((-1+i)^7\),after applying De Moivre's Theorem, we find:\[8\sqrt{2} \left(\cos(\frac{5\pi}{4}) + i \sin(\frac{5\pi}{4}) \right)\] which simplifies to \(-8 - 8i\).This demonstrates converting results back into rectangular form, confirming the calculations.

Complex Plane

The complex plane is a two-dimensional space to visualize complex numbers. It resembles the Cartesian coordinate system but adds a twist by expressing both real and imaginary components.

• The horizontal axis represents real numbers.
• The vertical axis represents imaginary numbers.

Each complex number corresponds to a unique point in this plane, often enriching understanding. For instance, the complex number \(-1+i\)equates to the point \((-1, 1)\).By visualizing this, we immediately see it lies in the second quadrant. This visualization becomes practical when calculating the argument, which is essential for converting numbers to polar form or applying De Moivre's Theorem.
Understanding the complex plane aids in grasping the geometry behind complex numbers. It influences interpretations in fields that rely on waveforms and oscillations, like engineering and physics.