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EBITDA
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Can √ 9 be √ (-3^2)? Updated on: 25 May 2016, 03:59
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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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.

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Can √ 9 be √ (-3^2)? 24 May 2016, 15:25
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EBITDA
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.
Show more
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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Can √ 9 be √ (-3^2)? 24 May 2016, 15:35
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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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Can √ 9 be √ (-3^2)? 25 May 2016, 01:40
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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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Can √ 9 be √ (-3^2)? 26 May 2016, 20:11
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EBITDA
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.
Show more

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#/2016/0 ... oots-gmat/
Here I discuss the reason why square root of 9 can only be 3.

Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!