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 now we have power number 48 in this we need to find find the value of z squared by w so z square

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.
$$
\frac{z^{2}}{w}
$$

Key Concepts

Complex Numbers

Complex numbers are mathematical objects that combine a real part and an imaginary part. They can be represented in the form a + bi, where a and b are real numbers and i is the square root of -1. This representation allows for the extension of real number arithmetic to include solutions to equations that would otherwise have no real solutions.

Rectangular and Polar Forms

A complex number can be expressed in rectangular form (a + bi) or in polar form, which uses a modulus (or absolute value) and an angle (or argument). The polar form offers a different perspective by describing the location of the point in the complex plane in terms of its distance from the origin and its direction relative to the positive real axis.

Modulus of a Complex Number

The modulus of a complex number is its distance from the origin in the complex plane and is calculated as the square root of the sum of the squares of its real and imaginary components. It provides a measure of the magnitude of the complex number and is an essential component when converting between rectangular and polar forms.

Argument of a Complex Number

The argument of a complex number is the angle that the line connecting the origin to the point makes with the positive real axis. This angle, usually measured in radians, indicates the direction of the complex number in the complex plane, and is used along with the modulus to express the number in polar form.

Principal Argument

The principal argument is the unique value of the argument that is conventionally taken within a specified range, typically between -? and ?. This standardization allows for consistency when comparing angles and performing operations involving complex numbers.

De Moivre's Theorem

De Moivre's Theorem states that when a complex number expressed in polar form is raised to an integer power, the modulus is raised to that power and the argument is multiplied by that integer. This theorem greatly simplifies the process of computing powers and roots of complex numbers.

Operations in Polar Form

When performing multiplication and division with complex numbers in polar form, the operation is performed on the moduli and the arguments separately. For multiplication, the moduli are multiplied and the angles are added; for division, the moduli are divided and the angle of the denominator is subtracted from the angle of the numerator. This method streamlines computations compared to performing the operations directly in rectangular form.

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