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charmedbeauty
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Homework Statement
z= 1+i√3
find z^9
Homework Equations
The Attempt at a Solution
Arg(z) = pi/3 and |z|=2
so z= 2e^i*pi/3
so z^9 = 2^9 (cos6pi +isin 6pi)
= 512(1) =512
but the answer has negative 512?
What is [itex]\displaystyle \left(2\,e^{i\pi/3}\right)^9\,?[/itex]charmedbeauty said:Homework Statement
z= 1+i√3
find z^9
Homework Equations
The Attempt at a Solution
Arg(z) = pi/3 and |z|=2
so z= 2e^i*pi/3
so z^9 = 2^9 (cos6pi +isin 6pi)
= 512(1) =512
but the answer has negative 512?
SammyS said:What is [itex]\displaystyle \left(2\,e^{i\pi/3}\right)^9\,?[/itex]
A complex number is a number that consists of both a real part and an imaginary part. It is written in the form a + bi, where a is the real part and bi is the imaginary part with the imaginary unit i.
To calculate the exponent of a complex number, you can use the formula (a + bi)^n = (a^n - b^n) + (na^(n-1)b)i, where n is the exponent and a and b are the real and imaginary parts, respectively.
Sure, let's say we want to calculate (3 + 2i)^3. Using the formula, we get (3^3 - 2^3) + (3(3^2)(2)i) = 27 - 8 + 18i = 19 + 18i.
Calculating the exponent of a complex number is important in many fields of science and engineering, such as in electrical engineering for solving circuits, in physics for calculating wave amplitudes, and in mathematics for solving differential equations.
Yes, there is a shortcut called De Moivre's formula, which states that (a + bi)^n = r^n(cos(nθ) + isin(nθ)), where r = sqrt(a^2 + b^2) and θ = tan^-1(b/a). This can be useful for calculating higher powers of complex numbers.