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A question about square roots

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A question about square roots [#permalink] New post 12 Jan 2009, 08:07
Can someone explain to me why:
sqrt4 = 2 (only positive 2)

while (x-1)^2= 16 equals:
x - 1 = 4
and
x - 1 = -4


is it because we don't know whether the value of x is positive or negative, thus we have to try out both scenarios?
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Re: A question about square roots [#permalink] New post 12 Jan 2009, 08:56
x-ALI-x wrote:
Can someone explain to me why:
sqrt4 = 2 (only positive 2)

while (x-1)^2= 16 equals:
x - 1 = 4
and
x - 1 = -4


is it because we don't know whether the value of x is positive or negative, thus we have to try out both scenarios?


1: sqrt4 = 2
sqrt4 is already a +ve. If it were to be -ve, it would be -(sqrt4).
Therefore, sqrt4 = 2.


2: Every variable has 2 roots: i.e. +ve and -ve. so (x-1)^2= 16 also has the same. i.e.
x - 1 = sqrt16 or -(sqrt16)

i: x - 1 = sqrt16
x - 1 = 4
x = 5

ii: x -1 = -(sqrt16)
x -1 = -(4)
x = -4 + 1
x = -3
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Re: A question about square roots [#permalink] New post 12 Jan 2009, 09:11
Excellent stuff Tiger, much obliged.
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Re: A question about square roots [#permalink] New post 12 Jan 2009, 09:35
agree with GT on the excellent explanation provided for 2.
I only want to add that:
sqrt(4)=+/-2 since squaring either 2 or -2 will yield 4 as the square.
So there is no confusion.

GMAT TIGER wrote:
x-ALI-x wrote:
Can someone explain to me why:
sqrt4 = 2 (only positive 2)

while (x-1)^2= 16 equals:
x - 1 = 4
and
x - 1 = -4


is it because we don't know whether the value of x is positive or negative, thus we have to try out both scenarios?


1: sqrt4 = 2
sqrt4 is already a +ve. If it were to be -ve, it would be -(sqrt4).
Therefore, sqrt4 = 2.


2: Every variable has 2 roots: i.e. +ve and -ve. so (x-1)^2= 16 also has the same. i.e.
x - 1 = sqrt16 or -(sqrt16)

i: x - 1 = sqrt16
x - 1 = 4
x = 5

ii: x -1 = -(sqrt16)
x -1 = -(4)
x = -4 + 1
x = -3

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Re: A question about square roots [#permalink] New post 12 Jan 2009, 10:31
tusharvk wrote:
agree with GT on the excellent explanation provided for 2.
I only want to add that:
sqrt(4)=+/-2 since squaring either 2 or -2 will yield 4 as the square.
So there is no confusion.


So this is exactly my confusion. From what I know sqrt(4) = +/-2
but in MGMAT's Number Properties book they say:
"Unlike even exponents, which yield both a positive and a negative solution, square roots have only one solution. For example:

If sqrt(4) = x, what is x?

In the above example, x = 2, since (2)(2) = 4. While it is true that (-2)(-2) = 4, the GMAT follows the standard convention that a radical (root) sign denotes only the non-negative root of a number. Thus, 2 is the only solution from x."
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Re: A question about square roots [#permalink] New post 12 Jan 2009, 10:48
tusharvk wrote:
agree with GT on the excellent explanation provided for 2.
I only want to add that:
sqrt(4)=+/-2 since squaring either 2 or -2 will yield 4 as the square.
So there is no confusion.


That is the issue: sqrt(4) is always 2 and not -2.
Lets simplify 4: 4 has two roots i.e. (i) sqrt (4) and (ii) -sqrt(4)

(i) sqrt (4) = 2
(ii) -sqrt(4) = -2.


hope it is clear.
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Re: A question about square roots [#permalink] New post 12 Jan 2009, 10:52
so here we are talking abt GMAc convention?
GMAT TIGER wrote:
tusharvk wrote:
agree with GT on the excellent explanation provided for 2.
I only want to add that:
sqrt(4)=+/-2 since squaring either 2 or -2 will yield 4 as the square.
So there is no confusion.


That is the issue: sqrt(4) is always 2 and not -2.
Lets simplify 4: 4 has two roots i.e. (i) sqrt (4) and (ii) -sqrt(4)

(i) sqrt (4) = 2
(ii) -sqrt(4) = -2.


hope it is clear.

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Re: A question about square roots [#permalink] New post 12 Jan 2009, 21:08
I do not think so.

tusharvk wrote:
so here we are talking abt GMAc convention?
GMAT TIGER wrote:
tusharvk wrote:
agree with GT on the excellent explanation provided for 2.
I only want to add that:
sqrt(4)=+/-2 since squaring either 2 or -2 will yield 4 as the square.
So there is no confusion.


That is the issue: sqrt(4) is always 2 and not -2.
Lets simplify 4: 4 has two roots i.e. (i) sqrt (4) and (ii) -sqrt(4)

(i) sqrt (4) = 2
(ii) -sqrt(4) = -2.


hope it is clear.


How is sqrt(4)=+/-2 true? Since sqrt(4) is +ve, so it should be 2 only.
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Re: A question about square roots [#permalink] New post 12 Jan 2009, 22:41
Here is my 3 cents on this topic:

1.Every positive number x has two square roots. One of them is positive and the other one is negative. If otherwise unqualified, "the square root" of a number refers to the principal square root, the non negative square root. In GMAT, the square root of any number is ONLY positive. We should follow MGMAT.

2. Can we have a sqrt of a negative number? Commonsense says no. Afterall, square of any root ( negative of positive) should be positive only. Then -2 should not have any root. Right?
Wrong! Square roots of negative numbers has been defined within the framework of complex numbers.This is done by introducing a new number, denoted by i (sometimes j) which is called the imaginary unit, which is defined such that (i) square = −1.

3.Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers.
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Re: A question about square roots [#permalink] New post 13 Jan 2009, 23:26
Here is one simple explanation. See if it helps.

Take n^2 = 4. n can be +2 or -2 . becos u can substitue both values and get the value
For n = 2 2^2 = 4.
For n = -2 -2^2 = 4.

Now,
Take sqrt(n) = 4.n can be +16 but not -16 . u can substitue both values and check
For sqrt(16) = 16^0.5 = 4.
For sqrt(-16) = complex number which u dont ve to bother in Gmat.

That is the reason , u dont consider -ve values when it comes to sqrt.
In simple words, Jus replace sqrt by ^0.5 , U ll be able to justify urself.
Hope this helped.

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Re: A question about square roots [#permalink] New post 14 Jan 2009, 00:59
There is a difference between talking about 'a square root of 16', and talking about \sqrt{16}. The square root ('radical') symbol means the non-negative square root. So while it's certainly true that 16 has two square roots, 4 and -4, if you ever see \sqrt{16}, this is always equal to 4 and only 4, because of the definition of the square root symbol.

And complex numbers are not tested on the GMAT- all numbers on the GMAT are real numbers, so for GMAT purposes, it is not permitted to take a square root of a negative.
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Re: A question about square roots [#permalink] New post 14 Jan 2009, 21:00
GMAT TIGER wrote:
x-ALI-x wrote:
Can someone explain to me why:
sqrt4 = 2 (only positive 2)

while (x-1)^2= 16 equals:
x - 1 = 4
and
x - 1 = -4


is it because we don't know whether the value of x is positive or negative, thus we have to try out both scenarios?


1: sqrt4 = 2
sqrt4 is already a +ve. If it were to be -ve, it would be -(sqrt4).
Therefore, sqrt4 = 2.


2: Every variable has 2 roots: i.e. +ve and -ve. so (x-1)^2= 16 also has the same. i.e.
x - 1 = sqrt16 or -(sqrt16)

i: x - 1 = sqrt16
x - 1 = 4
x = 5

ii: x -1 = -(sqrt16)
x -1 = -(4)
x = -4 + 1
x = -3


Here, I agree with GMAT Tiger. I just want to mention that, every "variable" has 2 roots, not numbers or integers. In your case, variable x=5 or x=-3 as GT mentioned. sqrt4=only 2, because the sqrt is positive by itself.
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Re: A question about square roots [#permalink] New post 16 Jan 2009, 13:50
x-ALI-x wrote:
Can someone explain to me why:
sqrt4 = 2 (only positive 2)

while (x-1)^2= 16 equals:
x - 1 = 4
and
x - 1 = -4


is it because we don't know whether the value of x is positive or negative, thus we have to try out both scenarios?


for further discussion: t27462
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Re: A question about square roots [#permalink] New post 17 Jan 2009, 17:15
GMAT TIGER wrote:
x-ALI-x wrote:
Can someone explain to me why:
sqrt4 = 2 (only positive 2)

while (x-1)^2= 16 equals:
x - 1 = 4
and
x - 1 = -4


is it because we don't know whether the value of x is positive or negative, thus we have to try out both scenarios?


for further discussion: t27462


Thanks GT, the thread you point to is good and weird :-).
Anyways, at this point, I am comfortable with what I have learned, primarily that sqrt4 is that of an integers, which is always positive, whereas (x-1)^2=16 involves a variable who's SIGN is unknown. Hence the latter gets the +/- outcomes.
right?

Thanks for the help everyone!
Re: A question about square roots   [#permalink] 17 Jan 2009, 17:15
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