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|.
See solution.
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. 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 a is even. Therefore, the expression will be real whether the value of a is positive, negative, or equal to 0.
In the radical, the index is even so the expression contained in the radical must be positive.
The product of b^3 and c^5 must be positive — b^3 and c^5 must have the same sign.
Then, b and c must have the same sign.
This means that we would need the absolute value symbols if we removed a, b, or 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.
Be aware that 3 and c^2 are non-negative numbers. Therefore, we can extract them from the absolute value symbol. Conversely, since a^3 and b can be either negative or positive, we should keep them in the absolute value symbol.
|3a^3bc^2|sqrt(2bc) ⇔ 3|a^3b|c^2sqrt(2bc)
