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Can √ 9 be √ (-3^2)?

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Manager
Joined: 24 May 2016
Posts: 129
Can √ 9 be √ (-3^2)?  [#permalink]

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Updated on: 25 May 2016, 02:59
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, 13:54.
Last edited by EBITDA on 25 May 2016, 02:59, edited 2 times in total.
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
Re: Can √ 9 be √ (-3^2)?  [#permalink]

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24 May 2016, 14:25
2
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.

Mike
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Mike McGarry
Magoosh Test Prep

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Re: Can √ 9 be √ (-3^2)?  [#permalink]

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24 May 2016, 14:35
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.
Manager
Joined: 24 May 2016
Posts: 129
Re: Can √ 9 be √ (-3^2)?  [#permalink]

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25 May 2016, 00:40
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!
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10490
Location: Pune, India
Re: Can √ 9 be √ (-3^2)?  [#permalink]

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26 May 2016, 19:11
1
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.

$$\sqrt{(-3)^2} = \sqrt{9}$$ 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: http://www.veritasprep.com/blog/2016/05 ... oots-gmat/
Here I discuss the reason why square root of 9 can only be 3.
_________________
Karishma
Veritas Prep GMAT Instructor

Re: Can √ 9 be √ (-3^2)?   [#permalink] 26 May 2016, 19:11