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by using de moivre's theorem but how?

2007-10-08 00:17:40 · 6 answers · asked by Anonymous in Science & Mathematics Mathematics

I've been trying to do this for hours!!! but I don't know how that's why I'm asking so if anyne can help me thank you.

2007-10-08 00:22:27 · update #1

6 answers

Well, the 4th root of 64 doesn't require de Moivre's theorem. Just remember that 64 = 2^6, so ∜64 = 2^(6/4) = 2^(3/2) = 2√2.

On the other hand, there was a poster earlier asking about the roots of the equation x⁴+64, so I suspect that you're asking how to find the 4th root of -64 (for which de Moivre's theorem would actually be useful):

First, note that the magnitude of -64 is 64 and it's argument is π. So we have -64 = 64 (cos (π+2πk) + i sin (π+2πk)), where k is any integer. So taking the fourth root, we take the fourth root of the magnitude and divide the argument by 4. So we have:

∜64 (cos (π/4 + kπ/2) + i sin (π/4 + kπ/2))

Now, we already know that ∜64 = 2√2. There are four distinct arguments the root can have: π/4, 3π/4, 5π/4, and 7π/4. Every other possible argument is coterminal to one of these. So the roots are:

2√2 (cos (π/4) + i sin (π/4)) = 2√2 (1/√2 + i/√2) = 2+2i
2√2 (cos (3π/4) + i sin (3π/4)) = 2√2 (-1/√2 + i/√2) = -2+2i
2√2 (cos (5π/4) + i sin (5π/4)) = 2√2 (-1/√2 - i/√2) = -2-2i
2√2 (cos (7π/4) + i sin (7π/4)) = 2√2 (1/√2 - i/√2) = 2-2i

Check:

(2+2i)⁴ = ((2+2i)²)² = (4 + 8i - 4)² = (8i)² = -64 ✓
(2+2i)⁴ = ((-2+2i)²)² = (4 - 8i - 4)² = (-8i)² = -64 ✓
(2+2i)⁴ = ((-2-2i)²)² = (4 + 8i - 4)² = (8i)² = -64 ✓
(2+2i)⁴ = ((2-2i)²)² = (4 - 8i - 4)² = (-8i)² = -64 ✓

So we are done.

2007-10-08 00:41:42 · answer #1 · answered by Pascal 7 · 1 0

Fourth Root Of 64

2017-01-09 07:02:20 · answer #2 · answered by ? 4 · 0 0

If you know how to get a square root, first do that and then again take the square root of that.

Square root of 64 is 8 by the way. And its square root is not an integer.

2007-10-08 00:26:21 · answer #3 · answered by Swamy 7 · 1 1

(but daveg^ ppl have problems and have to be guided when they dont know a particular thing i mean why do we have teachers than)

64 is not a perfect 4.i mean it does not have any no.that can be its 4 root.

eg 9 is not a perfect square.it does not have a proper square root.

2007-10-08 00:28:34 · answer #4 · answered by Anonymous · 0 3

why don't you use the calculator its simple. :) the answer is 2.828427125

2007-10-08 00:32:29 · answer #5 · answered by Anonymous · 0 3

poop

2014-10-09 13:26:52 · answer #6 · answered by Rupinder 1 · 1 0

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