How to Determine the Value of (-2 3i)4 in Polar Form

Introduction

In the realm of complex numbers, it is often necessary to work with numbers that are not in their standard form but are expressed in polar form. This article will guide you through the process of determining the value of (-2 3i)4 in polar form, a key operation in various applications of mathematics and engineering.

Understanding Complex Numbers

A complex number is a number that can be written in the form a bi, where a and b are real numbers, and i is the imaginary unit with the property that i2 -1. The number -2 3i is a complex number where the real part is -2 and the imaginary part is 3.

Conversion to Polar Form

The polar form of a complex number is given by r(cosθ isinθ), where r is the magnitude (or absolute value) and θ is the argument (or angle). To convert a complex number from rectangular form (a bi) to polar form, we need to find the values of r and θ.

The magnitude r is calculated as follows:

r √(a2 b2)

The angle θ is calculated as:

θ arctan(b/a) (adjusted for the correct quadrant)

For the complex number -2 3i, we can calculate:

r √((-2)2 (3)2) √(4 9) √13

To find θ, we need to consider the quadrant in which the complex number lies. Since the real part is negative and the imaginary part is positive, the number lies in the second quadrant. The angle is:

θ arctan(3/-2) (approximately -0.9828 radians, but adjusted for the second quadrant: θ π - 0.9828 ≈ 2.1628 radians)

Therefore, the polar form of -2 3i is:

(√13, 2.1628)

Raising to the 4th Power in Polar Form

Raising a complex number in polar form to a power is straightforward. If a complex number is in polar form (r, θ), then raising it to the power n results in:

(rn, nθ)

For (-2 3i)4, we have:

(r, θ) (√13, 2.1628)

n 4

Therefore, (r4, 4θ) (√134, 4 * 2.1628) (132, 8.6512) (169, 8.6512)

Converting Back to Rectangular Form

To convert the polar form back to rectangular form, we use the formula:

x rcosθ

y rsinθ

Here, x 169cos(8.6512) and y 169sin(8.6512). Calculating these:

x ≈ 169 * cos(8.6512) ≈ -172.29

y ≈ 169 * sin(8.6512) ≈ -113.77

Thus, the rectangular form of (-2 3i)4 is approximately -172.29 - 113.77i.

Conclusion

By following the steps outlined in this guide, you can easily determine the value of any complex number raised to a power in polar form. Whether you are dealing with intricate mathematical problems or real-world engineering applications, understanding polar form is a powerful tool.

Related Keywords

Complex numbers, polar form, scientific calculator, conversion, exponential form