%5E8.jpeg "(3-2i)^8")
Simplifying Complex Numbers: (3 - 2i)^8
This article aims to demonstrate the process of simplifying the expression (3 - 2i)^8, where 'i' represents the imaginary unit (√-1). We'll utilize the De Moivre's Theorem and polar form of complex numbers to efficiently solve this problem.
Understanding De Moivre's Theorem
De Moivre's Theorem states that for any complex number in polar form, z = r(cos θ + i sin θ) and any integer n, the following holds:
z^n = r^n (cos nθ + i sin nθ)
This theorem allows us to raise a complex number to a power by simply raising the modulus (r) to that power and multiplying the angle (θ) by the power.
Conversion to Polar Form
-
Finding the Modulus: The modulus of a complex number z = a + bi is given by |z| = √(a² + b²). For (3 - 2i), the modulus is: √(3² + (-2)²) = √(9 + 4) = √13
-
Finding the Angle: The angle (θ) is found using the arctangent function, considering the quadrant of the complex number. In this case: θ = arctan(-2/3) ≈ -33.69° (Since the real part is positive and the imaginary part is negative, it lies in the fourth quadrant)
Therefore, (3 - 2i) in polar form is: √13 (cos(-33.69°) + i sin(-33.69°))
Applying De Moivre's Theorem
Now, we can apply De Moivre's Theorem to simplify (3 - 2i)^8:
(3 - 2i)^8 = (√13)^8 [cos (8 * -33.69°) + i sin (8 * -33.69°)]
Simplifying further:
(3 - 2i)^8 = 13^4 [cos (-269.52°) + i sin (-269.52°)]
Since the cosine and sine functions have a period of 360°, we can simplify the angle:
(3 - 2i)^8 = 13^4 [cos (90.48°) + i sin (90.48°)]
Converting back to Rectangular Form
Finally, we can convert this back to rectangular form:
(3 - 2i)^8 = 13^4 * (cos 90.48° + i sin 90.48°) ≈ 13^4 * (0.039 + i * 0.999)
Therefore, (3 - 2i)^8 ≈ 28561 * (0.039 + 0.999i)
This gives us the simplified form of (3 - 2i)^8 in rectangular form.