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Can √ 9 be √ (-3^2)?

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EBITDA
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Can √ 9 be √ (-3^2)? [#permalink] New post   Updated on: 25 May 2016, 03:59
On a certain book I read that √ 9 can never be √ (-3^2) (**), because we can only put a number equal to or greater than 0 within a root of even index. Hence, √ 9 can only be √ (3^2).

However, on a different page of this same book, I also read the following:

Is it true that √X^2 = X?

Which develops as follows:

X=3 ---> √(3^2) = √9 = 3
X=-3 ---> √((-3)^2) = √9 = 3 (**)

Concluding that: √(X^2) = |X|

Mi question is: Don't notes marked with (**) contradict each other? Which one is correct?

May be I am missing something...

I would really appreciate that you would share your thoughts on this.

Thank you so much.

Originally posted by EBITDA on 24 May 2016, 14:54.
Last edited by EBITDA on 25 May 2016, 03:59, edited 2 times in total.
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Re: Can √ 9 be √ (-3^2)? [#permalink] New post   24 May 2016, 15:25
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EBITDA wrote:
On a certain book I read that √ 9 can never be √ (-3^2) (**), because we can only put a number equal to or greater than 0 within a root of even index. Hence, √ 9 can only be √ (3^2).

However, on a different page of this same book, I also read the following:

Is it true that √X^2 = X?

Which develops as follows:

X=3 ---> √(3^2) = √9 = 3
X=-3 ---> √(-3^2) = √9 = 3 (**)

Concluding that: √(X^2) = |X|

Mi question is: Don't notes marked with (**) contradict each other? Which one is correct?

May be I am missing something...

I would really appreciate that you would share your thoughts on this.

Thank you so much.

Dear EBITDA,
I'm happy to respond. :-) Many folks are confused on this particular point.

First of all, you may find this blog article helpful:
GMAT Quant: Roots

Here's the deal. We have to distinguish two cases that are easily confused.
1) Case #1: the √ appears printed on the page as part of the problem.
You see, technically, the √ symbol is the "find the positive square root" symbol. It always has a positive output (or a zero output), never a negative output. Thus, √9 equals +3 all the time.

2) Case #2: in the printed problem, a variable squared appears, and the student must take a square root in order to solve
In this suppose x^2 = 9 appears on the page: then we have to consider ALL roots, x = -3 and x = +3.

I'm not sure what the first book said, and in particular, I am not sure if you are aware of the subtleties of notation. Are you aware of the difference between these two:
(-3)^2 vs. -3^2
In the first, the squaring includes the negative, so the output is +9. In the second, by order of operations, we square the 3 first, then multiply by the negative, so the output is -9. We most certain cannot compute √ (-3^2) = √ (-9), because that is number not in the real number system. That's extremely different from
√ ((-3)^2) = √(9) = +3
I'm not sure whether the notational issue arose in the first book that you quoted or in your paraphrase of the first book.

It is definitely 100% true that
\(\sqrt{x^2} = |x|\)
That equation is true for every single number on the number line.

Does this answer your questions or not?
Mike :-)
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Re: Can √ 9 be √ (-3^2)? [#permalink] New post   26 May 2016, 20:11
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EBITDA wrote:
On a certain book I read that √ 9 can never be √ (-3^2) (**), because we can only put a number equal to or greater than 0 within a root of even index. Hence, √ 9 can only be √ (3^2).

However, on a different page of this same book, I also read the following:

Is it true that √X^2 = X?

Which develops as follows:

X=3 ---> √(3^2) = √9 = 3
X=-3 ---> √((-3)^2) = √9 = 3 (**)

Concluding that: √(X^2) = |X|

Mi question is: Don't notes marked with (**) contradict each other? Which one is correct?

May be I am missing something...

I would really appreciate that you would share your thoughts on this.

Thank you so much.


One thing I might add:

\(\sqrt{(-3)^2} = \sqrt{9}\) is fine.
The point is that when you take the square root of 9, that is only 3.
\(\sqrt{9} = 3\) only

You may find this post useful: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2016/05 ... oots-gmat/
Here I discuss the reason why square root of 9 can only be 3.
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Re: Can √ 9 be √ (-3^2)? [#permalink] New post   24 May 2016, 15:35
IMO, root of any number is only positive. \sqrt{9}= 3.

The second statement is saying the same thing. 9 = 3 * 3 or 9 = -3 * -3, irrespective of how you end up with 9, \sqrt{9} can only equal 3.
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Re: Can √ 9 be √ (-3^2)? [#permalink] New post   25 May 2016, 01:40
Dear Mike Mc Garry,

I have modified the 7th line of my original post, in which I omitted an important aspect (see text in blue).

X=-3 ---> √((-3)^2) = √9 = 3 (**)

Nonetheless, your response was very useful and cleared away all my doubts.

Thank you so much!
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Re: Can √ 9 be √ (-3^2)? [#permalink]
25 May 2016, 01:40
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