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# Square roots

Posted on 2011-05-03
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Can someone help me understand the sentence highlighted in blue. What is the root number? Thanks.

Are they referring to a tiny number that sometimes appears at the top just left of the radical sign? And if there is no number their it defaults to 2?
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Accepted Solution

>> What is the root number?
>> Are they referring to a tiny number that sometimes appears at the top just left of the radical sign? And if there is no root number, then it defaults to 2?

Yes
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Expert Comment

Notice that the root number, if not present, is, as you mentioned, 2.
Notation:

sqrt( y ) = y^(1/2)

So sqrt( y^2 ) = ( y^2 )^(1/2)

There is a rule:   (a^m) ^ n = a ^(mn)  when a > 0
Careful: In general, this is not the same as  a^(m ^ n)

So, (a^m) ^ (1/n) = a ^( m*(1/n) ) = a^( m/n )
and if m = n, then (a^n) ^ (1/n) = a^(n/n) = a^1 = a

But if a < 0, and if n is even, then if y < 0, we want the n-th root of y^n to be positive. So, to be safe, the absolute value signs are used to handle y either postive or negative.
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Expert Comment

"The reason we use the absolute value is ...."          wrong      if the number is even

If a number is raised to an even power the result is always > 0
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Author Closing Comment

Thanks to both of you for your help.
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LVL 26

Expert Comment

actually I guess they are right if you define simplify in a certain way, which is probably what they are doing
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Author Comment

phoffric sent me this site link a couple of weeks ago. I am finding it helpful.

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