Rational Exponents and Radicals

For any real number $a,$ the radical expression $\sqrt[n]{a^n}$ can be simplified as follows. $$\begin{array}{c}\sqrt[n]{a^n}=\{\begin{array}{l}a\text{ if }n\text{ is odd}\\ |a|\text{ if }n\text{ is even}\end{array}\end{array}$$ 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. $$\begin{array}{c}\sqrt{64a{b}^4{c}^5}\end{array}$$ 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 $8b^2{c}^2$ is always positive. Therefore, we don't need absolute value symbols. $$\begin{array}{c}\left|8b^2{c}^2\right|\sqrt{ac}\iff 8b^2{c}^2\sqrt{ac}\end{array}$$