In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 + i)⁵

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Recall DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)th power is given by \(r^n (\cos(n\theta) + i \sin(n\theta))\).

Convert the complex number \(1 + i\) into polar form by finding its magnitude \(r\) and argument \(\theta\). The magnitude is \(r = \sqrt{1^2 + 1^2}\) and the argument is \(\theta = \arctan\left(\frac{1}{1}\right)\).

Apply DeMoivre's Theorem to raise the complex number to the 5th power: compute \(r^5\) and multiply the angle by 5 to get \(5\theta\). So the result in polar form is \(r^5 (\cos(5\theta) + i \sin(5\theta))\).

Convert the result back to rectangular form by evaluating \(r^5 \cos(5\theta)\) for the real part and \(r^5 \sin(5\theta)\) for the imaginary part.

Write the final answer as \(a + bi\), where \(a\) and \(b\) are the real and imaginary parts found in the previous step.

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주요 개념

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DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number expressed in polar form as r(cos θ + i sin θ), its nth power is given by r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.

Conversion between Rectangular and Polar Forms

Complex numbers can be represented in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting to polar form involves finding the magnitude r = √(a² + b²) and the argument θ = arctan(b/a), which is essential for applying DeMoivre's Theorem.

Rectangular Form of Complex Numbers

Rectangular form expresses complex numbers as a + bi, where a is the real part and b is the imaginary part. After using DeMoivre's Theorem in polar form, the result is converted back to rectangular form by evaluating cos nθ and sin nθ to find the real and imaginary components.