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Math Help - Negative number raised to the two-thirds power

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    Negative number raised to the two-thirds power

    Hello. I'm a calculus 1 student in college, and I recently came across a function where x was raised to the (2/3) power. Does anyone in this forum know why a negative number raised to the 2/3 becomes an imaginary number? Type the problem into any calculator, including Google. You'll get an error or an imaginary number.

    (-1)^(2/3)

    But why does this give you an imaginary number? It's the same as ((-1)^2)^(1/3) or ((-1)^(1/3))^(2). It seems like when you square (-1) you get (1), and the cube root of (-1) is just (-1). Can anyone on this forum explain why this happens?
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    Re: Negative number raised to the two-thirds power

    Quote Originally Posted by cristumagnus View Post
    Hello. I'm a calculus 1 student in college, and I recently came across a function where x was raised to the (2/3) power. Does anyone in this forum know why a negative number raised to the 2/3 becomes an imaginary number? Type the problem into any calculator, including Google. You'll get an error or an imaginary number.

    (-1)^(2/3)

    But why does this give you an imaginary number? It's the same as ((-1)^2)^(1/3) or ((-1)^(1/3))^(2). It seems like when you square (-1) you get (1), and the cube root of (-1) is just (-1). Can anyone on this forum explain why this happens?
    It is because of the way the calculator compute non-integer powers (essentially taking logs, which requires that the number being raised to the power be positive, also if you calculator is smart enough to use complex logarithms it won't return the answer you expect since the branch used is not the real branch).

    CB
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    MHF Contributor alexmahone's Avatar
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    Re: Negative number raised to the two-thirds power

    Quote Originally Posted by cristumagnus View Post
    Hello. I'm a calculus 1 student in college, and I recently came across a function where x was raised to the (2/3) power. Does anyone in this forum know why a negative number raised to the 2/3 becomes an imaginary number? Type the problem into any calculator, including Google. You'll get an error or an imaginary number.

    (-1)^(2/3)

    But why does this give you an imaginary number? It's the same as ((-1)^2)^(1/3) or ((-1)^(1/3))^(2). It seems like when you square (-1) you get (1), and the cube root of (-1) is just (-1). Can anyone on this forum explain why this happens?
    (-1)^{2/3}

    =(\cos \pi+i\sin \pi)^{2/3}

    =\cos \frac{2}{3}\pi+i\sin \frac{2}{3}\pi

    =-0.5 + 0.866025404 i
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    Re: Negative number raised to the two-thirds power

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    Re: Negative number raised to the two-thirds power

    Quote Originally Posted by alexmahone View Post
    (-1)^{2/3}

    =(\cos \pi+i\sin \pi)^{2/3}

    =\cos \frac{2}{3}\pi+i\sin \frac{2}{3}\pi

    =-0.5 + 0.866025404 i
    And how did you determine which branch to use?

    CB
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    Re: Negative number raised to the two-thirds power

    Ti-84 ...
    Attached Thumbnails Attached Thumbnails Negative number raised to the two-thirds power-twothirdspower.jpg  
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Re: Negative number raised to the two-thirds power

    Quote Originally Posted by CaptainBlack View Post
    And how did you determine which branch to use?

    CB
    I suppose there are many ways of calculating (-1)^{2/3}.

    Another would be:

    (-1)^{2/3}

    =(\cos 3\pi+i\sin 3\pi)^{2/3}

    =\cos \left(\frac{2}{3}\times 3\pi\right)+i\sin \left(\frac{2}{3}\times 3\pi\right)

    =\cos 2\pi+i\sin 2\pi

    =1
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    Re: Negative number raised to the two-thirds power

    Thank you. This is the answer I got from Google. Can you tell me what math subject I can study to learn about this? I want to build my math skills. I will find books and videos if you tell me the specific subject. For example, in college algebra and trigonometry, we never studied any of this. I have become very interested in math and I want to study math.
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    Re: Negative number raised to the two-thirds power

    Quote Originally Posted by cristumagnus View Post
    Thank you. This is the answer I got from Google. Can you tell me what math subject I can study to learn about this? I want to build my math skills. I will find books and videos if you tell me the specific subject. For example, in college algebra and trigonometry, we never studied any of this. I have become very interested in math and I want to study math.
    Complex numbers.
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  10. #10
    MHF Contributor alexmahone's Avatar
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    Re: Negative number raised to the two-thirds power

    To sum up, all 3 cube roots of unity are valid answers for (-1)^{2/3}.
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