Exercise 48 Page 420

For any real number $a,$ the radical expression $\sqrt[n]{a^n}$ can be simplified as follows. 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 possible values of the variables $a,$ $b,$ and $c.$

• In the radical, the index is even and the exponents of $a$ and $b$ are odd. Therefore, in order for this radical expression to result in a real number, $a$ and $b$ must be non-negative.
• In the radical, the index is even and the exponent of $c$ is even. Therefore, the expression will be real whether the value of $c$ is positive, negative or equal to $0.$

This means that if we remove $a$ or $b$ from the radical, we will not need absolute value symbols. However, we would need them if we removed $c$ from the radical. We can simplify the radical by writing the expression inside as powers with exponents equal to the index of the radical using the Product Property of Radicals. The simplest form of the expression is $2a^2{c}^4\sqrt[4]{3a{b}^3}.$