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Question
Fourth roots of 625i
a) find the indicated roots of the complex number.
b) represent each of the roots graphically
c) express each of the roots in standard form.
Thank you!
Questions will be answered at the end of this note.
Read on to find out how to do it.
Look at the answer and scan through what was written to find the answers.
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To find the 4th root of a complex number, convert it to polar coordinates.
The real positive numbers are at 0°, and the positive irrational numbers are at 90°.
That is what we have is a number with n = 625 and α = 90°.
To find the 4th root, first you need to know that 5*5*5*5=625,
so 5 is the 4th root of the number.
For the angle 90°, since we are finding the fourth root, get four angles.
The angles are the starting angle with 360° added to it 3 times.
That gives α1 = 90°, α2 = 450°, α3 = 810°, and α4 = 1170°.
The reason we added 360° to the angle 0, 1, 2, and 3 times
is that now we divide those angles by 4.
That gives α1 = 18.5°, α2 = 112.5°, α3 = 202.5°, and α4 = 292.5°.
So the four roots are the real numbers with magnitude 5 at each of these angles.
If you want these numbers converted back to cartesian coordinates (x-y),
x = 5cos(αi) and y = 5sin(αi) for i = 1, 2, 3, and 4.
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Answer on how to do questions:
a) Once they are converted back to (x,y), they are in standard form.
b) To draw them graphically, take a piece of x,y graph paper with (0,0) in the center.
Remember that x is at 0° and y is at 90°. Note also the -x is at 180° and -y is at 270°.
Plot them on a circle of radius 5 at the proper angle.
c) They are, in standard form,
(5·cos(18.5°), 5·sin(18.5)°),
(5·cos(112.5°), 5·sin(112.5)°),
and very similar formulas with the angle 810° and 1170°.
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