Find each power. Write answers in rectangular form. [3 cis ...

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, typically written in the form a + bi. They can be represented in different forms, including rectangular form, which shows the components directly, and polar form, which encapsulates magnitude and direction.

Polar and Rectangular Forms

Polar form expresses a complex number in terms of its magnitude (or modulus) and angle (or argument), while rectangular form expresses it as a sum of a real part and an imaginary part. Understanding these representations is crucial because different operations are easier in different forms; for example, exponentiation is more straightforward in polar form, whereas addition and subtraction are simpler in rectangular form.

cis Notation

The cis notation is a shorthand for representing a complex number in polar form, where cis ? is defined as cos ? + i sin ?. This notation simplifies expressions and calculations involving trigonometric functions when dealing with complex numbers.

De Moivre's Theorem

De Moivre's theorem is a key principle in complex number theory that facilitates the process of raising complex numbers in polar form to integer powers. It states that for a complex number expressed as r cis ?, raising it to the nth power results in r^n cis (n?), making it easier to compute powers and convert the result back to rectangular form.

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