find nth root of complex number z for the specified value of n : z = 4(squareroot of 3) - 4i, n=3?

Question

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z= 4( _ / 3 ) - 4i, n=3
i have tried my level best to explain 4 multiplied by squareroot of 3 & then rest is obvious

Answer 1

z = 4√3 - 4i
=4 (√3 - i)
=4*2(√(3)/2 - ½i) (on multiplying and dividing by √ ((√3)² + 1²))
=8(cos(π/6 + 2kπ) - isin(π/6 + 2kπ))
=8(cos(11π/6 + 2kπ) + isin(11π/6 + 2kπ)) (because cos angle is >0 and sin angle < 0 therefore angle in 4th quadrant)
=8e^(i11π/6 + 2kπ)

z^⅓ = (2e^(i11π/6 + 2kπ))^⅓
=8^⅓.e^(i11π/18 + 2kπ/3)
=2(cos(11π/18 + 2kπ/3)) + isin(11π/18 + 2kπ/3))

So the distinct cube roots are:
2(cos(11π/18 + 0)) + isin(11π/18 + 0)),
2(cos(11π/18 + 2π/3)) + isin(11π/18 + 2π/3)) and
2(cos(11π/18 + 4π/3)) + isin(11π/18 + 4π/3))

ie, on simlpifying, 2(cos(11π/18)) + isin(11π/18)),
2(cos(23π/18)) + isin(23π/18)) and
2(cos(35π/18)) + isin(35π/18))

Yeah Actuator I did ... carelessness is my sad downfall!! But NOW its correct Guaranteed!! ;^))

Answer 2

Wal C has the right idea, but he made at least one error and got the wrong answer. Perhaps the following will be understandable to you.

To repeat the question:
You're asking for the cube root (because n = 3) of
4 * sqrt (3) - 4 i

The methodology for taking roots of complex numbers (or finding the complex roots of real numbers) is as follows:

1. Plot the number on the complex plane (i.e., on a graph, where the "x" axis represents real numbers and the "y" axis represents imaginary numbers). In this case, you'll plot a point that is
4 * sqrt (3) units to the right of the "y" axis, and
4 units below the "x" axis.

2. Determine the magnitude of a line segment from the origin to the plotted point. In this case, we have a right triangle with two sides of length 4 * sqrt (3) and 4. By Pythagoras, the magnitude is 8 units.

3. Take the (real, positive) nth root of the magnitude as determined in step 2.
The cube root of 8 is 2. this is the magnitude of the cube root of the complex number.

4. Determine the angle of the line segment from the origin to the plotted point. The angle is determined as in trigonometry: You start at 0 degrees on the "x" axis, and go counterclockwise until you reach the point. The positive "y" axis is 90 degrees; the negative "x" axis is 180 degrees; the negative "y" axis is 270 degrees.
And because the triangle discussed in step 2 is a 30-60 right triangle, the line segment in question is at an angle of 330 degrees.

5. Divide the angle found in step 4 by n.
330 / 3 = 110 degrees.

6. The answer to the problem has a magnitude equal to the value found in step 3, at an angle equal to the angle found in step 5:
2 units and 110 degrees. But you need to convert this to the usual complex number form. The method for this is:
magnitude * cos (angle) + i * magnitude * sin (angle)
In this case, it is
2 cos 110 deg + 2 i sin 110 deg

7. When you take the nth root of a number, there are n nth roots. They all have the same magnitude, and are distributed symmetrically around the 360 degrees about the origin. Since this problem involves a cube root, there are 3 values, and they are 120 degrees (= 360 / n) apart. So you just take the answer in step 6 and add 120 to the 110 degrees (= 230 degrees), and then add 120 again (= 350 degrees) to get the other two roots.

Answer 3

First time period, multiply good and bottom by technique of (3 - 4i). this gives you z(3 - 4i)/(3 - 4i)(3 + 4i) = z(3 - 4i)/25 = (3z - 4iz)/25 2d time period, multiply good and bottom by technique of -i. this gives you -i(z - a million)/-i*5i = -iz + i/5 0.33 time period, multiply good and bottom by technique of (3 + 4i). this gives you 5(3 + 4i)/(3 + 4i)(3 - 4i) = 5(3 + 4i)/25 = (3 + 4i)/5 putting those at the same time you get (3z - 4iz)/25 + (-iz + i)/5 = (3 + 4i)/5 Multiply entire equation by technique of 25 to get 3z - 4iz -5iz + 5i = 15 + 20i 3z - 9iz = 15 + 15i z - 3iz = 5 + 5i z(a million - 3i) = 5 + 5i z (a million - 3i)(a million + 3i) = (5 + 5i)(a million + 3i) 10z = -10 + 20i z = -a million + 2i Why did not one of the others fact on the actual shown actuality that their solutions were not the only given by technique of your e book?

Answer 4

The easiest way to do this is to convert the cartesian complex number to a polar form. I'll post a link for you to read (I prefer to tell people how to do it rather than just give them the answer). Mathworld is an excellent source of math info. Ah, too late, two people already said this. Oh well, at least I know I'm right.

Answer 5

symbols would be nicer if possible this is kind of incoherent sorry i can't help.