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.

_________________

PhD in Applied Mathematics

Love GMAT Quant questions and running.