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.
|a^3 |sqrt(6)/9
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 even
Since 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 variable a in the numerator.
sqrt(8a^6)/sqrt(108)
In the radical, the index is even and the exponent of a is even. Therefore, the expression will be real whether the value of a is positive or negative. Remember that we will need to use an absolute value expression in this case.
Next, we need to rationalize the denominator. We can do that by multiplying our fraction by a fraction equivalent to 1 such that the resulting denominator is a rational number. Remember that we have to multiply both the numerator and denominator.
