A Square Root Has Only One Value

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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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 !!!

Hi sguntaka,

Thanks for the reply.
Just wanted to know why this is notion is different from what we are taught in school/colleges?

Any reasons?

Thanks,
FKA

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: -√

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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