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. $$\left[\sqrt{3}\left(\cos 70^{\circ}+i \sin 70^{\circ}\right)\right]^{3}$$$$\left[2^{1 / 5}\left(\cos 63^{\circ}+i \sin 63^{\circ}\right)\right]^{10}$$

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.

$$\left[\sqrt{3}\left(\cos 70^{\circ}+i \sin 70^{\circ}\right)\right]^{3}$$$$\left[2^{1 / 5}\left(\cos 63^{\circ}+i \sin 63^{\circ}\right)\right]^{10}$$

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Key Concepts

Conversion between Polar and Rectangular Forms

This concept involves converting a complex number from its polar representation to the rectangular form, a + bi, where a and b are real numbers. The conversion uses the formulas a = r cos ? and b = r sin ?, and it is essential when expressing results in a more familiar algebraic form, especially after performing operations using De Moivre's Theorem.

Complex Numbers in Polar Form

This concept refers to expressing a complex number in the form r(cos ? + i sin ?), where r is the magnitude (or modulus) of the complex number and ? (theta) is the argument or angle. This representation is especially useful for multiplication, division, and exponentiation of complex numbers because it easily separates the scaling factor from the rotation in the complex plane.

De Moivre's Theorem

De Moivre's Theorem provides a formula to raise complex numbers in polar form to integer powers. It states that [r(cos ? + i sin ?)]^n = r^n (cos(n?) + i sin(n?)), which simplifies the process of exponentiation by scaling the modulus and multiplying the angle. This theorem is central to many problems involving powers and roots of complex numbers.

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