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    Why is it that these have different solutions ?
    x^2=16
    x = √16
    Why is it that the first one has 2 solutions and the second one had only one. My confusion arises since the second equation is just the first one square rooted.
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    The second equation defines x to be the (positive) square root of 16.

    The first equation defines x to be any value which, when squared, will equal 16. The positive and negative root of 16 satisfies the first equation.
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    (Original post by MangoFreak)
    The first equation defines x to be any value which, when squared, will equal 16.
    .. But so does the second one..
    I kind of understand you... but the second equation says that x is any number that when timed by itself equals 16, doesn't it? So that includes -4.
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    What made you ask this question, out of interest?
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    (Original post by Noble.)
    What made you ask this question, out of interest?
    I don't understand it...
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    It only makes sense if you do the principal square root.
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    (Original post by MathMeister)
    I don't understand it...
    Did it just pop into your head, or did it arise while problem solving? I ask because your problem is almost a non-problem, the only reason the second equation can be considered to only have one solution is because often the \sqrt{} sign is taken to mean "take the principal square root", but it's by no means unambiguous.
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    (Original post by Noble.)
    Did it just pop into your head, or did it arise while problem solving? I ask because your problem is almost a non-problem, the only reason the second equation can be considered to only have one solution is because often the \sqrt{} sign is taken to mean "take the principal square root", but it's by no means unambiguous.
    It is rather important to know the difference, as in an exam, if a question asks you to square root something and you need to find 2 values, you may get it wrong and only find one.
    Thank you , as well. You have made me understand.
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    (Original post by MathMeister)
    It is rather important to know the difference, as in an exam, if a question asks you to square root something and you need to find 2 values, you may get it wrong and only find one.
    Obviously yes, however I think depending on the situation it is normally clear what is required because I personally can't think of a time (either from A-Level maths to undergrad) where I've not known whether they want the principal solution, or all solutions.
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    (Original post by Noble.)
    Obviously yes, however I think depending on the situation it is normally clear what is required because I personally can't think of a time (either from A-Level maths to undergrad) where I've not known whether they want the principal solution, or all solutions.
    Have you got examples?
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    (Original post by MathMeister)
    Have you got examples?
    If you have an equation that involves x^2 then there will be 2 solutions (perhaps repeated)


    I am intrigued by your OP

    x = \sqrt{16} is not an equation as such so it does not have solution(s)

    It is a bit like asking how many solutions x = 5 has


    Also it is not "the first one square rooted"

    It you take the square root of both sides of the original equation you get

    x^2 = 16

    x = \pm\sqrt{16}
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    (Original post by TenOfThem)
    If you have an equation that involves x^2 then there will be 2 solutions (perhaps repeated)


    I am intrigued by your OP

    x = \sqrt{16} is not an equation as such so it does not have solution(s)

    It is a bit like asking how many solutions x = 5 has


    Also it is not "the first one square rooted"

    It you take the square root of both sides of the original equation you get

    x^2 = 16

    x = \pm\sqrt{16}
    Thanks, it all makes perfect sense now.
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    (Original post by Noble.)
    often the \sqrt{} sign is taken to mean "take the principal square root", but it's by no means unambiguous.
    Often? Isn't it always? Is there ever a scenario where \sqrt{x} is interpreted as anything other than +\sqrt{x}? (at least in the modern usage - maybe 100 years ago there was some ambiguity)
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    (Original post by TenOfThem)
    x = \sqrt{16} is not an equation as such so it does not have solution(s)
    I'd have to disagree: it's a complete standard equation, as it asserts that one quantity is equal to another quantity, and it most certainly does have a solution (for example, x=1 is not a solution since 1 \neq \sqrt{16}, but x = \sqrt{16} is a solution since \sqrt{16} = \sqrt{16}).

    To put it another way, if you were asked to write down the solution set of that expression, what set would you write down?

    It is a bit like asking how many solutions x = 5 has
    That may or may not be a trivial question. In modular arithmetic, that equation can have an infinite number of solutions.
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    (Original post by atsruser)
    I'd have to disagree: it's a complete standard equation, as it asserts that one quantity is equal to another quantity, and it most certainly does have a solution (for example, x=1 is not a solution since 1 \neq \sqrt{16}, but x = \sqrt{16} is a solution since \sqrt{16} = \sqrt{16}).

    To put it another way, if you were asked to write down the solution set of that expression, what set would you write down?



    That may or may not be a trivial question. In modular arithmetic, that equation can have an infinite number of solutions.
    Hence my use of "as such"
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    (Original post by atsruser)
    Often? Isn't it always? Is there ever a scenario where \sqrt{x} is interpreted as anything other than +\sqrt{x}? (at least in the modern usage - maybe 100 years ago there was some ambiguity)
    It probably is, I know I've seen that convention used in at least one textbook but I don't actually know how universally accepted it is.
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    I did a mini-study on square roots/sqrt symbol etc. a few months ago and I was amazed by how many experienced maths teachers taught things like \sqrt{4}=\pm 2, which confuses students when they see \sqrt{x} referring to the principal square root in A-Level textbooks.
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    (Original post by Noble.)
    It probably is, I know I've seen that convention used in at least one textbook but I don't actually know how universally accepted it is.
    It's pretty much universal now that both \sqrt{x} and x^{1/2} refer to the positive square root of x when x is a positive real number.

    I suspect it has been the case for a significant period of time, otherwise the plus-or-minus symbol used in the standard quadratic solution formula would be redundant, and I learned that formula over 30 years ago!

    It still doesn't stop textbooks being in error, though - I've seen examples where a student is asked to simplify something innocent-looking, like (x^2y^4)^{1/2} and the "answer" is given as xy^2 without the required modulus sign around the x.
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    (Original post by atsruser)
    Often? Isn't it always? Is there ever a scenario where \sqrt{x} is interpreted as anything other than +\sqrt{x}? (at least in the modern usage - maybe 100 years ago there was some ambiguity)
    If x is a positive number I agree with you, but you might still see questions like "what is the Riemann surface of  \sqrt{z(z-1)(z-2)} ?" in other contexts.
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    (Original post by RichE)
    If x is a positive number I agree with you, but you might still see questions like "what is the Riemann surface of  \sqrt{z(z-1)(z-2)} ?" in other contexts.
    I'm afraid you're being too subtle for me here. Isn't a function single-valued on its Riemann surface, in which case I'm not sure what ambiguity can arise. Or have I forgotten too much complex analysis?
 
 
 
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