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.
2x^2 ysqrt(11y)
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 non-negative. Otherwise, the radical would be imaginary. With this in mind, let's consider the possible values of the variables, x and y.
In the radical, the index is even and the exponent of x is even. Therefore, the expression will be real for all real values of x.
In the radical, the index is even and the exponent of y is odd. Therefore, in order for this radical expression to result in a real number, y must be non-negative.
With this information, we can simplify the expression.
Now, let's consider the absolute value expression, |2x^2y|. We have to determine whether or not 2x^2y is greater than or equal to 0. We will do it for 2, x^2, and y, one at a time.
The number 2 is greater than 0.
Every real number squared is greater than or equal to 0, so x^2 ≥ 0.
Recall that y has to be non-negative, in order for this radical expression to result in a real number.
Summing up, the expression 2x^2y is greater than or equal to 0, since it is a product of three non-negative terms. Thus, we can remove the absolute value.
