Guest said:If $z = e^{(2-\frac{i \pi}{4})}$ what's $z^5$?
The only way I can think of doing this is expanding $(2-\frac{i \pi}{4})^5$, but I think I'm supposed to use a simpler method (not sure what it's).
Thank you, I like Serena. I get it now. (Smile)I like Serena said:Hi Guest, (Smile)
Let's substitute and apply a power rule:
$$z^5=\left(e^{(2-\frac{i \pi}{4})}\right)^5
=e^{5(2-\frac{i \pi}{4})}
$$
Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part is represented by the letter "i" and is defined as the square root of -1. Complex numbers are written in the form a + bi, where a is the real part and bi is the imaginary part.
To simplify complex numbers, you can use the properties of square roots and combine like terms. First, simplify the real and imaginary parts separately. Then, combine the simplified parts using addition or subtraction, depending on the sign between the two terms.
There are a few rules to follow when simplifying complex numbers. First, you can only combine terms that have the same imaginary part. Second, when multiplying complex numbers, use the FOIL method (multiply the First, Outer, Inner, and Last terms). Lastly, when dividing complex numbers, use the conjugate of the denominator to rationalize the fraction.
Yes, complex numbers can have decimals or fractions in their real and imaginary parts. For example, 3.5 + 2.25i is a complex number with decimal parts, and 1/2 + 3/4i is a complex number with fractional parts. The same rules for simplifying complex numbers apply, regardless of the form of the numbers.
Complex numbers are essential in math and science because they allow us to solve problems that cannot be solved with real numbers. For example, complex numbers are used in electrical engineering to represent AC circuits, in quantum mechanics to describe the behavior of particles, and in signal processing to analyze and manipulate signals.