Question: Root x Squared

Comment on Root x Squared

In the previous video, it is shown that sqroot of 49 can not be -7. why does sqroot of 16 make -4?
gmat-admin's picture

This is really an issue about notation. So, first of all, √16 does not equal -4. √16 equals 4 only.
From the Official Guide for GMAT Review: A square root of a number n is a number that, when squared, is equal to n. Every positive number n has two square roots, one positive and the other negative, but √n (i.e., the square root NOTATION) denotes the POSITIVE number whose square is n.
For example, √9 denotes 3. The two square roots of 9 are √9 = 3 and –√9 = –3.

Hi,
It is somewhat confusing. Could you please explain it using another example? It would be much appreciated.
gmat-admin's picture

Sure thing.
If x² = 25, then we know that x = 5 or x = -5
We might says that x = the square root of 25.
So, there are two square roots of 25 (5 and -5)

However, the square root NOTATION (√) directs us to single out the POSITIVE square root of a value.

So, for example, √25 = 5, √36 = 6, √49 = 7, etc.

Hi,
Does that mean the solution shown in the video is wrong and both the statements are sufficient ?

gmat-admin's picture

The solution provided in the video is correct. Statement 1 is not sufficient, and statement 2 is sufficient.

It all boils down to notation.

I think the confusion on this might be that in the video, they mention:

if x^2 = 16, x has two roots: -4 and 4.

However, if it was written √16 in the GMAT, then they are only referring to the POSITIVE root: 4.

Does that makes sense?

This video series is great. I am finding it very helpful. Thank you,
Sugumar
gmat-admin's picture

Ah, I see.

The "root" of an equation is the same as the solution. So, for example, we might ask "What is the root of the equation 3x = 6?"
Here, the root is x = 2

Likewise, when I say that the equation x² = 16 has two roots (x = -4 and x = 4), I'm referring to the solutions to the equation.

How about the rule that states that the √of x^2 equals the |x|.Wouldn't this mean then that the first statement is sufficient in proving x= +4
gmat-admin's picture

Not quite.

You're right in that √(x²) = |x|

So, in statement 1, we can replace √(x²) with |x| to get: |x| = 4

Now, if |x| = 4, what is the value of x?

Well, x can have TWO POSSIBLE values. Either x = 4 or x = -4. So, statement 1 is still not sufficient.

Cheers,
Brent

Hi Brent,
thank you for the great tutorial.
But I have one more question: A root can also be written as ^(1/2). So when I rewrite (1), I have the equation x^2^(1/2)=4, which is equal to x=4. How can I figure out the -4 with this approach?
Best
Eric
gmat-admin's picture

That approach, although somewhat valid, prevents us from recognizing that, in the original equation, the part INSIDE the root (in statement 1) must be calculated FIRST.

So, if we square either 4 or -4, we get 16, and THEN we find the square root of 16.

Hey. I'm having a hard time understanding how the second one is sufficient. The way I see it, the first statement could have x = 4 or x = - 4, so we don't know which one it is, and therefore it's not sufficient.

With statement 2, isn't it the same? X could be 2 or it could be - 2. So how is this sufficient, but the first statement isn't? Hmm..
gmat-admin's picture

Hi Alovald,

This question tests an important feature of square root notation.

You might want to read some of the discussions above. The basic idea here is that the square root NOTATION (√) directs us to single out the POSITIVE square root of a value.

So, for example, √4 = 2 (not -2)

So, for statement 2, when we reach the conclusion that √x = 2 or √x = -2, we can ELIMINATE the possibility that √x = -2.

Does that help?

Cheers,
Brent

Hi Brent,

firstly - great videos and question sets. With respect to the question posed in the video, if we were to solve it mathematical - wouldn't the following be true?

statement 1:

sqrt (x^2) = 4
this can be re-written as:

(x^2) ^ (1/2) = 4

x^2/2 = 4

x = 4

therefore, we get one value for x. I'm trying to figure out why this is incorrect - can you help please.
gmat-admin's picture

Great question!!!

We must take into consideration that, by performing your simplification, we end up disregarding the fact that, in the ORIGINAL equation, 4² = 16 and (-4)² = 16. In other words, squaring a positive and squaring a negative value both result in a POSITIVE number .

To be 100% safe, we must say that √(x²) = |x| (not just x)

More here: https://math.stackexchange.com/questions/59630/square-root-of-a-number-s...

TAKEAWAY: Before simplifying an expression, consider whether there are any ramifications of doing so.

Cheers,
Brent

Thanks for the prompt response. Very clear!

Hi Brent! In respect of the second statement why can' we simply state that the square of any number will be positive thus the only possible value is positive?
gmat-admin's picture

Be careful. That strategy won't work in many cases.

EXAMPLE: If we're told that x² = 9, we can't conclude that x = 3
If x² = 9, then EITHER x = 3, since 3² = 9
OR x = -3, since (-3)² = 9

Cheers,
Brent

Hello Brent, All,

The question asks "What is the value of x," but it doesn't clarify whether to 1) give the value of x before the order of operations, or 2) give the value of x during/after the order of operations. The answer given in the video is the correct answer in the paradigm of point #2.

Is this not a valid observation?

However if we use that aforementioned quote from the Official Guide, should we always assume in any GMAT question that the sqrt of any number n will always be positive, even before calculation?

Regards,

Eric
gmat-admin's picture

The order of operations is imposed on us by the notation.

For example, if we must calculate something like √(7²), we must calculate the value of 7² before we can take the square root.

Likewise, if we must calculate something like (√9)², we must calculate the value of √9 before we can square the value.

The same applies to √(x²) and (√x)²

Does that help?

Cheers,
Brent

This question truly throws one for a loop :) Since it's a given that statement 2 is a factual statement, then we would have to assume x = 4 given the equation itself does not account for it to be a "no solution." If we did stubbornly assume regardless that x=-4 prior to calculation, then it wouldn't be a factual statement, as the left side equals positive 4, as per the equation itself. That was the boundary that I had missed. So in reality, it wasn't about a question of calculation or pre-calculation, but rather a stress test on the equation itself to determine if there's only one answer. Thank you!
gmat-admin's picture

Great analysis!

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