one fourth root of w is 2+3j what are the rest

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pepsi

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hello again,

and thanks in advance for any hint/help

the moduli of the root i am given is 13^1/2 so the moduli of w is (13^1/2 )^4

what I am having difficulty with is finding the argument of w. the argument of the root i am given is arctan(3/2) = 56,309

and the forumla linking this with the argument of w is 56,309 = (2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.

how can I progres from here. please help.

x
 
pepsi said:
hello again,

and thanks in advance for any hint/help

the moduli of the root i am given is 13^1/2 so the moduli of w is (13^1/2 )^4

what I am having difficulty with is finding the argument of w. the argument of the root i am given is arctan(3/2) = 56,309

and the forumla linking this with the argument of w is 56,309 = (2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.

how can I progres from here. please help.

x
Click to expand...

What is the EXACT question?
 
one fourth root of w is 2+3j. Find w and its other fourth roots, and represent all five points on an Argand diagram
 
What is j? A complex number? Also, use decimal points not comas: 56.309 not 56,309.
 
pepsi said:
hello again,

the moduli of the root i am given is 13^1/2 so the moduli of w is (13^1/2 )^4

what I am having difficulty with is finding the argument of w. the argument of the root i am given is arctan(3/2) = 56,309

and the forumla linking this with the argument of w is 56,309 = (2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.

how can I progres from here. please help.

x
Click to expand...
Try drawing the picture in the first quadrant for 2 + 3i [Sorry, but only you engineering types use 'j'.]

Draw your right triangle with x = 2, y = 3, r = sqrt(13).

Next rotate that 90 degrees into Quad II, and draw the (similar) triangle. I think you'll find (-3,2) is the next root.

Continue.
 
PAULK said:
pepsi said:
hello again,

the moduli of the root i am given is 13^1/2 so the moduli of w is (13^1/2 )^4

what I am having difficulty with is finding the argument of w. the argument of the root i am given is arctan(3/2) = 56,309

and the forumla linking this with the argument of w is 56,309 = (2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.

how can I progres from here. please help.

x
Click to expand...
Try drawing the picture in the first quadrant for 2 + 3i [Sorry, but only you engineering types use 'j'.]

Draw your right triangle with x = 2, y = 3, r = sqrt(13).

Remember that the fourth roots will be spaced 90 degrees around the circle.

So rotate that 90 degrees into Quad II, and draw the (similar) triangle. I think you'll find (-3,2) is the next root.

Continue.
Click to expand...
 
One way to find 'w'

w = (2 + i3)[sup:39tr94ye]4[/sup:39tr94ye] = [(2 + i3)[sup:39tr94ye]2[/sup:39tr94ye]][sup:39tr94ye]2[/sup:39tr94ye] = [4 + i12 - 9][sup:39tr94ye]2[/sup:39tr94ye] = [i12 - 5][sup:39tr94ye]2[/sup:39tr94ye] = 25 - i120 - 144 = -119 - i120

So 'w' is in third quadrant with magnitude 13[sup:39tr94ye]2[/sup:39tr94ye] = 169
 
pepsi said:
hello again,

and thanks in advance for any hint/help

the moduli of the root i am given is 13^1/2 so the moduli of w is (13^1/2 )^4

what I am having difficulty with is finding the argument of w. the argument of the root i am given is arctan(3/2) = 56,309

and the forumla linking this with the argument of w is 56,309 = (2kpi + arg(w))/4, which is true for a specific value of k out of the possible values of k, i.e. 0,1,2,3.

how can I progres from here. please help.

x
Click to expand...

You don't need to work out w.

The 'useful fact' to use here is that the n roots of a complex number are equally spaced around a circle centred at the origin (0+0i). So the fourth roots of w will be four points on a circle around 0, with equally-spaced arguments.

PS - arctan(3/2) is not 56.309 - you should forget about degrees, and only use radians from now on, for 99.99% of the maths you will face in the future.
 
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