SOLUTION: find all the roots and express you answers in standard/rectangular form. 2. (-64)^1/4

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Question 1178752: find all the roots and express you answers in standard/rectangular form.
2. (-64)^1/4
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

%28-64%29%5E%281%2F4%29
=root%284%2C-64%29
=root%284%2C-1%2A64%29
=root%284%2C64%29%2Aroot%284%2C-1%29..........root%284%2C64%29+=2sqrt%284%2C4%29=2sqrt%284%2C2%5E2%29=2sqrt%282%29,
2sqrt%282%29root%284%2C-1%29->result

or, in decimal form:
since root%284%2C-1%29=%281+%2B+i%29%2Fsqrt%282%29, we have
=2sqrt%282%29%28%281+%2B+i%29%2Fsqrt%282%29%29
=2%281+%2B+i%29
=2+%2B+2i

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


You should know how to express -64 as a complex number, in trigonometric form, and in exponential form:

-64+=+-64%2B0i+=+64%28cis%28pi%29%29+=+64e%5E%28i%2Api%29

To find the "primary" 4th root, use deMoivre's Theorem with the trigonometric form, or, equivalently, use the exponential form. I will show using deMoivre's Theorem -- to take the 4th root, take the 4th root of the magnitude, and divide the angle by 4:


Convert that to rectangular form: the angle is pi/4 (45 degrees), and the magnitude is 2sqrt(2). That gives us (-64)^(1/4) = (2+2i).

That root is easily verified:

%282%2B2i%29%5E4+=+%28%282%2B2i%29%5E2%29%5E2+=+%288i%29%5E2+=+-64

To find the other 4th roots of -64, use the result of deMoivre's Theorem that says the four 4th roots of any complex number have the same magnitude and are spaced around the complex plane at intervals of 360/4 = 90 degrees.

That gives us the four 4th roots of -64 in rectangular form:

2+2i
-2+2i
-2-2i
2-2i