- #1

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[tex](-16)^{1/4}=2i^{1/4}[/tex]

[tex]i=e^{i \pi/2}[/tex]

[tex]i^{1/4}=e^{i \pi/8}[/tex]

[tex](-16)^{1/4}=2e^{i \pi/8}[/tex]

or

[tex](-16)^{1/4}=2e^{i 5\pi/8}[/tex]

or

[tex](-16)^{1/4}=2e^{i 9\pi/8}[/tex]

is this correct?

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- Thread starter UrbanXrisis
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- #1

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[tex](-16)^{1/4}=2i^{1/4}[/tex]

[tex]i=e^{i \pi/2}[/tex]

[tex]i^{1/4}=e^{i \pi/8}[/tex]

[tex](-16)^{1/4}=2e^{i \pi/8}[/tex]

or

[tex](-16)^{1/4}=2e^{i 5\pi/8}[/tex]

or

[tex](-16)^{1/4}=2e^{i 9\pi/8}[/tex]

is this correct?

- #2

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As far as it goes. Where is your fourth 4th root?

- #3

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the fourth root of -16 is [tex]2e^{i 9\pi/8}[/tex]

didnt i show that?

didnt i show that?

- #4

Hurkyl

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- #5

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- #6

Hurkyl

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In the original question, you said you're supposed to find the 4th rootI am to find the 4th root of -16

I guess I'm not up on the convention for this stuff, but I would say that this is wrong. I would say the L.H.S. is multivalued, and denotes[tex](-16)^{1/4}=2e^{i 9\pi/8}[/tex]

- #7

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I see, so they are equal but just not equal in showing ALL the fourth roots of -16 right?

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- #9

Hurkyl

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I agree. Lots of silly mistakes are made because people forget that the inverse of the

- #10

HallsofIvy

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What??? "they are all equal but just not equal"???UrbanXrisis said:I see, so they are equal but just not equal in showing ALL the fourth roots of -16 right?

Any number has 4 distinct fourth (complex) roots. For example the fourth roots of 1 are 1, -1, i, and -i. You were asked to find all of the fourth roots of -16. ("i am to find the 4th root

You only showed three in your original post.

Actually, your very first statement:

[tex](-16)^{1/4}=2i^{1/4}[/tex]

is wrong. The principle root of 16 is, of course, 2 but -1 is not equal to i!

What you should have written was

[tex](-16)^{1/4}= 2(-1)^{1/4}[/tex]

Now, what are the 4 distinct fourth roots of -1?

- #11

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[tex](-1)^{1/4}=e^{9i \pi /4}[/tex]

[tex](-1)^{1/4}=e^{17i \pi /4}[/tex]

[tex](-1)^{1/4}=e^{25i \pi /4}[/tex]

right? so that:

[tex](-16)^{1/4}= 2e^{i \pi /4}[/tex]

or

[tex](-16)^{1/4}= 2e^{9i \pi /4}[/tex]

or

[tex](-16)^{1/4}= 2e^{17i \pi /4}[/tex]

or

[tex](-16)^{1/4}= 2e^{25i \pi /4}[/tex]

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