Complex constant from single root.

In summary, we are asked to determine the complex constant c for which the given equation has a root of v = √3 - j. After analyzing the equation and using the formula rejθ = r(cosθ + jsinθ), we can express v as an exponential and solve for c.
  • #1
DmytriE
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Homework Statement


Do not use a calculator for this problem. Express your answers using square roots and/or fractional multiples of x.

Determine the complex constant c such that v is a root of: z6 - c = 0


Homework Equations


v = [itex]\sqrt{3} - j[/itex]


The Attempt at a Solution



I believe the following are true:
1. There are 3 distinct roots for this equation.
2. Each of the distinct roots will have a conjugate pair to ensure there is no middle term.
3. If 2 is true, then two of the six roots are [itex]\sqrt{3} -j[/itex] and [itex]\sqrt{3} +j[/itex]

z6 - c = 0 --> z6 = c. Yes, this is as far as I have gotten. I'm not sure how to figure out the other 4 roots.
 
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  • #2
c = z6
v = √3 - j
Use rejθ = r(cosθ + jsinθ)
Write v as exponential
Solve for c
 

Related to Complex constant from single root.

1. What is a complex constant from single root?

A complex constant from single root is a numerical value that can be represented as a combination of a real number and an imaginary number. It is derived from a single root, which is the solution to a complex equation.

2. How is a complex constant from single root calculated?

A complex constant from single root is calculated by taking the square root of the negative value inside the radical sign in the complex equation. This value is then multiplied by the complex conjugate of the denominator to eliminate the imaginary component.

3. What is the significance of a complex constant from single root in science?

In science, complex constants from single root are used to represent physical quantities that have both magnitude and direction, such as electric and magnetic fields. They are also used in signal processing, control systems, and other fields of engineering.

4. Can a complex constant from single root have a negative value?

Yes, a complex constant from single root can have a negative value. This indicates that the imaginary component is pointing in the opposite direction of the real component, and the overall value is in the fourth quadrant of the complex plane.

5. How is a complex constant from single root different from a real constant?

A complex constant from single root is different from a real constant because it includes an imaginary component, while a real constant only has a real component. This means that a complex constant from single root can have both magnitude and direction, while a real constant only has magnitude.

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