%5E5-in-exponential-form.jpeg "(1-i)^5 In Exponential Form")
Simplifying (1-i)^5 in Exponential Form
This article explores the process of simplifying the complex number (1-i)^5 into its exponential form.
Understanding Exponential Form
The exponential form of a complex number is a way to express it in terms of its magnitude (or modulus) and angle (or argument). This form is particularly useful for performing operations like multiplication and division of complex numbers. It is expressed as:
z = r * e^(iθ)
Where:
- z is the complex number
- r is the magnitude of the complex number
- θ is the angle (in radians) of the complex number
- e is the base of the natural logarithm (approximately 2.71828)
- i is the imaginary unit (√-1)
Simplifying (1-i)^5
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Finding the magnitude (r) and angle (θ) of (1-i):
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Magnitude (r): The magnitude of a complex number (a+bi) is calculated as √(a² + b²).
Therefore, for (1-i), the magnitude is √(1² + (-1)²) = √2. -
Angle (θ): The angle of a complex number is calculated using the arctangent function (tan⁻¹). The angle for (1-i) is tan⁻¹(-1/1) = -π/4 radians (or -45°). Since (1-i) lies in the fourth quadrant, we need to add 2π to the angle to get the principal angle: -π/4 + 2π = 7π/4 radians.
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Applying De Moivre's Theorem: De Moivre's Theorem states that for any complex number in polar form (r * e^(iθ)) and any integer n:
(r * e^(iθ))^n = r^n * e^(inθ)
Therefore, to calculate (1-i)^5, we can apply De Moivre's Theorem:
(1-i)^5 = (√2 * e^(i7π/4))^5 = (√2)^5 * e^(i35π/4)
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Simplifying the result:
- (√2)^5 = 4√2
- 35π/4 can be simplified to 7π/4 (by subtracting multiples of 2π).
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Final exponential form:
Therefore, (1-i)^5 in exponential form is 4√2 * e^(i7π/4).
Conclusion
By converting (1-i) to exponential form and applying De Moivre's Theorem, we successfully simplified (1-i)^5 to its exponential form. This method is often more efficient for dealing with powers of complex numbers, especially when dealing with larger exponents.