Carry out the indicated operations. Express your results in...

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator.

$$\left[3\left(\cos \frac{1}{3} \pi+i \sin \frac{1}{3} \pi\right)\right]^{5}$$

Key Concepts

Rectangular Form

Rectangular form is the expression of a complex number as a + bi, which depicts the real and imaginary parts separately. Converting from polar to rectangular form often involves evaluating trigonometric functions to obtain the real and imaginary components explicitly.

De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form, (r(cos ? + i sin ?))^n equals r^n (cos(n?) + i sin(n?)). This theorem simplifies the process of raising complex numbers to integer powers by operating directly on the magnitude and angle.

Trigonometric Functions

Trigonometric functions like cosine and sine are used to relate angles to ratios of sides in right triangles. In the context of complex numbers, they convert angular information from polar form into the horizontal (real) and vertical (imaginary) components of the number.

Complex Numbers

Complex numbers are numbers expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1). They provide a systematic way to work with numbers that have both real and imaginary parts, making them fundamental in many areas of mathematics.

Polar Form

Polar form represents a complex number using its magnitude (modulus) and angle (argument) in the form r(cos ? + i sin ?). This representation is particularly useful for performing operations such as multiplication, division, and raising numbers to a power.

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