For any real number a contained in a radical such that sqrt(a^n), the root is a if n is odd and |a| if n is even.
12ab^2sqrt(ab)
Practice makes perfect
To simplify radicals, we should recall the rules regarding when the root should be contained inside an absolute value. Consider the following two cases for any real number a.
sqrt(a^n)=
a if n is odd
|a| if n is evenSince the radical is a real number and the root is even, the expression underneath the radical is positive. Otherwise, the radical would be imaginary. With this in mind, let's consider the possible values of the variables, a and b.
The index is even and the exponents of a and b are odd. For the expression to result in a real number, the product of a and b must be positive. Therefore, a and b must have the same sign — both positive or both negative.
In this case, we will not need an absolute value. We can now simplify the given radical by writing the expression inside as powers equal to the index number of the radical.
