For any real numbera, the radical expressionnan can be simplified as follows.
nan={∣a∣ifnisodd∣a∣ifniseven
Since the radical is a real number and the index of the root is even, the expression underneath the radical is positive. Otherwise, the radical would be imaginary. 64ab4c5
With this in mind, let's consider the possible values of the variables, a,b, and c.
In the radical, the index is even and the exponent of b is even. Therefore, the expression will be real whether the value of b is positive, negative or equal to 0.
In the radical, the index is even and the exponents of a and c are odd. Since b4 is always positive, in order for this radical expression to result in a real number, the product of a and c5 must be also positive — a and c5 must have the same sign.
This means that if we remove a,b, and c from the radical, we will need absolute value symbols.