If n is an odd number, then the radical expression sqrt(a^n) simplifies to a. If n is even, the expression sqrt(a^n) simplifies to |a|.
6b^2c^2 sqrt(ac)
Practice makes perfect
For any real number a, the radical expression sqrt(a^n) can be simplified as follows.
sqrt(a^n)=
a if n is odd
|a| if n is even
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.
sqrt(36ab^4c^5)
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 b^4 is always positive, in order for this radical expression to result in a real number, the product of a and c^5 must be also positive — a and c^5 must have the same sign.
This means that if we remove a, b, and c from the radical, we will need absolute value symbols.
Notice that both b and c have even exponents, so 6b^2c^2 is always positive. Therefore, we do not need absolute value symbols.
|6b^2c^2| sqrt(ac) ⇔ 6b^2c^2 sqrt(ac)
