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Simplifying Complex Numbers: (1 + i)^(3/4) + 3i
This article will guide you through simplifying the complex number (1 + i)^(3/4) + 3i. We will use the properties of complex numbers and polar form to achieve the solution.
Understanding Complex Numbers
A complex number is a number of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (i² = -1).
De Moivre's Theorem
De Moivre's Theorem provides a powerful tool for simplifying powers of complex numbers. It states that for any complex number z = r(cos θ + i sin θ) and any integer n, the following holds true:
zⁿ = rⁿ(cos (nθ) + i sin (nθ))
Simplifying (1 + i)^(3/4)
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Convert to Polar Form: We need to express (1 + i) in polar form, which involves finding its magnitude (r) and angle (θ).
- Magnitude: r = √(1² + 1²) = √2
- Angle: θ = arctan(1/1) = π/4 (Since (1 + i) lies in the first quadrant)
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Apply De Moivre's Theorem: Now we have (1 + i) = √2(cos(π/4) + i sin(π/4)). To find (1 + i)^(3/4), we apply De Moivre's theorem:
- (1 + i)^(3/4) = (√2)^(3/4) * (cos((3/4) * (π/4)) + i sin((3/4) * (π/4)))
- (1 + i)^(3/4) = 2^(3/8) * (cos(3π/16) + i sin(3π/16))
Adding 3i
Finally, we add 3i to the simplified expression:
(1 + i)^(3/4) + 3i = 2^(3/8) * (cos(3π/16) + i sin(3π/16)) + 3i
Expressing the Final Result
The final answer can be expressed in the following forms:
- Polar Form: 2^(3/8) * (cos(3π/16) + i sin(3π/16)) + 3i
- Rectangular Form: You can convert the polar form into rectangular form by calculating the cosine and sine values and then adding the real and imaginary components.
Conclusion
Simplifying complex expressions like (1 + i)^(3/4) + 3i requires applying the fundamental properties of complex numbers, specifically De Moivre's Theorem. Understanding these concepts allows us to manipulate and simplify complex expressions efficiently.