Use DeMoivre's Theorem to find the indicated power of the...

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form.
$$\left[\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)\right]^{6}$$

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Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number represents it in terms of its magnitude (or modulus) and angle (or argument). This form, expressed as r(cos ? + i sin ?), is especially useful in power and root calculations since multiplication and division of complex numbers can be performed easily by operating on the magnitudes and angles.

Rectangular (Cartesian) Form of Complex Numbers

The rectangular form expresses a complex number as a + bi, where a and b are the real and imaginary parts, respectively. Converting a complex number from its polar form to its rectangular form involves calculating the cosine and sine of the angle multiplied by the magnitude, thus providing an alternative and often more intuitive format for interpreting and graphing the number.

De Moivre's Theorem

De Moivre's Theorem is a fundamental result in complex number theory that provides a formula to raise complex numbers to a power when they are expressed in polar form. The theorem states that for a complex number expressed as r(cos ? + i sin ?), its nth power is given by r^n (cos n? + i sin n?), which simplifies operations like exponentiation and root extraction.

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