For Exercises 41-52, use z=-(3 √(3))/(2)+(3)/(2) i and w=3 √(2)-3 i √(2) to compute the quantity

step 1 we have probably number 45 in this we need to find the value of w cube so w in polar form is 6 s

For Exercises 41-52, use z=-(3 √(3))/(2)+(3)/(2) i and w=3...

For Exercises $41-52,$ use $z=-\frac{3 \sqrt{3}}{2}+\frac{3}{2} i$ and $w=3 \sqrt{2}-3 i \sqrt{2}$ to compute the quantity, Express your answers in polar form using the principal argument.
$$
w^{3}
$$

Key Concepts

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form, (r e^(i?))^n = r^n e^(i n?), where n is an integer. This theorem is fundamental for computing powers and roots of complex numbers, making it easier to handle problems involving exponentiation in the complex plane.

Modulus and Argument

The modulus of a complex number, given by ?(a² + b²), measures the distance from the origin in the complex plane, while the argument, usually measured in radians, indicates the angle made with the positive real axis. These components are critical when converting a complex number from rectangular to polar form.

Principal Argument

The principal argument is the unique representation of the angle associated with a complex number, typically chosen to lie within a specific interval, such as (-?, ?] or [0, 2?). It standardizes the angular part of the polar form ensuring consistency when expressing results.

Complex Numbers

Complex numbers are expressions of the form a + bi, where a and b are real numbers and i is the imaginary unit that satisfies i² = -1. They extend the real number system and are essential in various fields of mathematics, physics, and engineering.

Polar Form of Complex Numbers

The polar form represents a complex number in terms of its modulus (or absolute value) and its argument (or angle), and is typically written as r(cos ? + i sin ?) or equivalently re^(i?). This form simplifies multiplication, division, and the process of raising numbers to integer powers.

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