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Thread: square root

  1. #1 square root 
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    Hi,

    I have trouble making calculations with square root. When the square root is normal I can easily play with it to make a better number (eg: √32 = 2√8 )
    But I get stuck when I get something like this: ^4√32 (32^1/4)
    So when the exponent of the square root changes like the latter example I get confused. Hopefully someone can explain me this.
    Thanks


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  3. #2 Re: square root 
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    Quote Originally Posted by gamelogic
    Hi,

    I have trouble making calculations with square root. When the square root is normal I can easily play with it to make a better number (eg: √32 = 2√8 )
    But I get stuck when I get something like this: ^4√32 (32^1/4)
    So when the exponent of the square root changes like the latter example I get confused. Hopefully someone can explain me this.
    Thanks
    Your statement is a little confusing. Are you asking if ^4√32 = (32^1/4) ?

    If so, it is a simple matter of notation. In general nth root if anything is the same as 1/nth power. To understand the equivalence, raise both to the nth power.


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  4. #3 Re: square root 
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    [quote="mathman"]
    Quote Originally Posted by gamelogic

    Your statement is a little confusing. Are you asking if ^4√32 = (32^1/4) ?

    If so, it is a simple matter of notation. In general nth root if anything is the same as 1/nth power. To understand the equivalence, raise both to the nth power.
    Firstly thanks for your reply.

    With this: ^4√32 = (32^1/4) I meant this ^4√32 (32^1/4). My mistake.

    But the Problem that I have is how can I finish this ^4√32.
    The finished answer is 2*^4√(2) But I just don't see how to work these things out. The number ^4 on the square root messes up my ability to finish these things.
    The same problem with these: ^3√(X) and ^5√(X) or ^6√(X) etc...

    With normal √(X) I can reason how to work to the finished form but with the latter
    square roots I just can't reason well. What is the trick to finish those square roots.
    Maybe if I see some examples I can understand it probably but an explanation would be great.

    So in short, I can do this: √(40) = √(4*10) = 2√(10)

    But when I see this I don't know how to work: ^4√(40)

    I want to know what's the technique there.
    Thanks
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  5. #4  
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  6. #5  
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    Quote Originally Posted by JaneBennet
    Yes, thanks. I think I am beginning to understand it. It looks like
    you just worked out the inside first. And then you went to the outside.
    But still one little thing. Why did you do 2*16 in step 2 and not 4*8. Is it a rule
    to get the smallest number in the square root with the option to put one
    of the numbers out of the square root?

    And by the way really thanks, I was making my homework yesterday with a slowmotion rate of one math question per hour just because of this different square root :? .
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  7. #6  
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    Quote Originally Posted by gamelogic
    Quote Originally Posted by JaneBennet
    Yes, thanks. I think I am beginning to understand it. It looks like
    you just worked out the inside first. And then you went to the outside.
    But still one little thing. Why did you do 2*16 in step 2 and not 4*8. Is it a rule
    to get the smallest number in the square root with the option to put one
    of the numbers out of the square root?

    And by the way really thanks, I was making my homework yesterday with a slowmotion rate of one math question per hour just because of this different square root :? .
    One of the possibly confusing things lies in the conventions of the symbols.

    for instance, we tend to write numbers as

    1, 2, 3, 4 etc

    when, for algebraic purposes, the y are treated as +1, +2, +3, +4 etc.

    Similarly, the root sign you used arose conventionally for square roots but, technically speaking, you are better off understanding it by replacing the bare sign with an explicit one.

    That is instead of saying:



    try saying



    This will help you understand and appreciate how your sign works: it tells you how many roots of the number in question you have to take.

    Thus



    means to find a number that, if multiplied by itself twice (2 times), makes 4.

    And



    represents a number that, if multplied by itself 5 times, makes 32.

    The other way of representing this, in more consistent format with our use of indices, is to show them as numbers being raised to fractional powers.

    So



    is exactly the same as saying

    4<sup>1/2</sup>

    And



    is exactly the same as saying

    32<sup>1/5</sup>

    This latter method of representing these numbers works particularly well because it reminds us of the close relationship between roots and powers of numbers, and allows us to think of roots as another form of power (that is, a number 'raised to' a certain power).

    I don't know if this helps, but if you were struggling with notation/conceptualisation, it might.

    cheer

    shanks
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  8. #7  
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    Actually, you could also do it using fractional exponents:

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  9. #8  
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    Thanks for the good explanation. You have helped me a lot!
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  10. #9  
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    Quote Originally Posted by JaneBennet
    Actually, you could also do it using fractional exponents:

    This is completely correct, and an efficient way to do the problem

    What never ceases to baffle me, however, is that beyond a good exercise in writing the same idea down in different way, one of which may be more useful than the others in some particular context, the concept that somehow is "simpler" than . It seems to me that the distinction is rather artificial without some overriding purpose to guide the "simplification".

    In short, I think you have produced an elegant solution to a silly problem. Replacing one relativtly mysterious quantity with a second mysterious quantity is a bit pointless It is these kinds of silly problems that I think give neophytes a very distorted view of what mathematics is all about. .
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