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# A Square Root Has Only One Value

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Intern
Joined: 27 May 2012
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GMAT Date: 05-17-2012
A Square Root Has Only One Value [#permalink]

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17 Jun 2012, 11:56
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Hi all,

I am going through the Manhattan Series and found that the even root (a square root, a 4th root, a 6th root, etc.), a radical sign means ONLY the non negative root of a number.
Can anyone please elaborate on this? And also suggest the similar counterpart for the odd root.

Thanks,
FKA

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Re: A Square Root Has Only One Value [#permalink]

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17 Jun 2012, 12:36
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When the GMAT provides the square root sign for an even root, such as (square root x) or (4th root x), then the only accepted answer is the positive root

Example, square root 25 = 5, NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, +5 and -5
Even roots have only a positive value on the GMAT.

Odd roots will have the same sign as the base of the root.
For example, cube root 125 = 5 and cube root -64 =-4

Hope this is clear. Thanks !!!

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Re: A Square Root Has Only One Value [#permalink]

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17 Jun 2012, 12:55
Hi sguntaka,

Just wanted to know why this is notion is different from what we are taught in school/colleges?

Any reasons?

Thanks,
FKA

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Re: A Square Root Has Only One Value [#permalink]

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17 Jun 2012, 14:21
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MyFutureMyProspects wrote:
Hi all,

I am going through the Manhattan Series and found that the even root (a square root, a 4th root, a 6th root, etc.), a radical sign means ONLY the non negative root of a number.
Can anyone please elaborate on this? And also suggest the similar counterpart for the odd root.

Thanks,
FKA

Roots are functions for which we use designated symbols. Finding a root of a number means finding the value of a function for a specific value of the variable.
Square root is a function. A function can return just one unique value. It was chosen, by definition, that all even order roots, return the positive value. For odd order roots, there is no problem, the answer is always unique.

Examples:
Square root of 25 means - I am a function, you gave me the number 25 (this is x) I return you another number y (and just one number) which squared equals 25.
It is true that -5 squared is also 25, but if I am a function, I cannot return two different values. So, by definition, the positive value was chosen and square root of 25 is 5, not -5.

No problems with cubic roots. Cubic root of 8 is simply 2, cubic root of -27 is -3. There are no other possibilities. No negative number raised to an odd power gives positive number, and no positive number raised to an odd power gives a negative number.

Equations can have many solutions. Solving an equation can involve computations of some roots. So, if $$x^2=25$$, then $$x=\sqr{25}=5$$, and also $$x=-\sqr{25}=-5$$, because both, when squared, give 25. Also, we can rewrite the equation as $$x^2-25=0$$ or $$(x-5)(x+5)=0$$. A product is 0 when one of the factors is 0. So, either $$x-5=0$$ or $$x+5=0$$, and we find the same two solutions.
_________________

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Intern
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Re: A Square Root Has Only One Value [#permalink]

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17 Jun 2012, 20:45
The radical sign ( √ ) always means the positive (principal) square root.
Thus, if both the positve and negative roots are needed, then: +-√ is used.
If you want to signify the negative root only, then that would be: -√

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Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
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Location: Pune, India
Re: A Square Root Has Only One Value [#permalink]

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19 Jun 2012, 23:15
MyFutureMyProspects wrote:
Hi all,

I am going through the Manhattan Series and found that the even root (a square root, a 4th root, a 6th root, etc.), a radical sign means ONLY the non negative root of a number.
Can anyone please elaborate on this? And also suggest the similar counterpart for the odd root.

Thanks,
FKA

It's right.

$$x^2 = 16$$ has two solutions: x = 4 or -4

$$x = \sqrt{16}$$ has only one solution: x = 4
'the square root' is used to refer to only the positive square root.
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Re: A Square Root Has Only One Value [#permalink]

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10 Jun 2014, 08:54
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Re: A Square Root Has Only One Value [#permalink]

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01 Mar 2016, 17:38
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: A Square Root Has Only One Value   [#permalink] 01 Mar 2016, 17:38
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