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: RootsHere'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.
-3^2In 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
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)