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 problem number 46 and sorry problem 47 we need to find z cube w square k so we have polar form o

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.
$$
z^{3} w^{2}
$$

Key Concepts

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. They extend the real numbers and include an imaginary part, which allows for the solution of equations that have no real solutions, and provide a way to represent quantities in the complex plane.

Polar Form

The polar form of a complex number expresses it in terms of a magnitude (or modulus) and an angle (or argument). It is written as r(cos ? + i sin ?) or equivalently re^(i?), which often simplifies the multiplication, division, and exponentiation of complex numbers.

Rectangular to Polar Conversion

Converting a complex number from rectangular form (a + bi) to polar form involves computing the modulus r = sqrt(a^2 + b^2) and the argument ? = arctan(b/a), taking into account the sign of a and b to determine the correct quadrant. This conversion is essential when leveraging the multiplicative properties of complex numbers in polar form.

Principal Argument

The principal argument is the unique value of the argument of a complex number chosen within a specified interval, usually (–?, ?] or [0, 2?). This standardization avoids ambiguity in the representation of angles and is crucial when expressing final results in polar form.

Euler's Formula

Euler's formula states that e^(i?) is equivalent to cos ? + i sin ?. This relationship provides a powerful link between exponential functions and trigonometric functions, simplifying the manipulation and computation of complex numbers in exponential or polar forms.

De Moivre's Theorem

De Moivre's Theorem is used to raise complex numbers in polar form to integer powers. It states that (r e^(i?))^n = r^n e^(i n?), which significantly simplifies problems involving exponentiation of complex numbers by operating directly on the modulus and argument.

Multiplication in Polar Form

Multiplying complex numbers in polar form is straightforward because the moduli multiply and the arguments add. If two complex numbers are written as r1e^(i?1) and r2e^(i?2), their product is (r1r2)e^(i(?1+?2)). This property is fundamental in combining multiple complex factors, especially when working with powers and products.

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