The expression **radical expression.** The symbol **radical sign.** The expression under the radical sign is called the **radicand,** and *n*, an integer greater than 1, is called the **index.** If the radical expression appears without an index, the index is assumed to be 2. The expression *n*th root of *a*.” Remember:

Example 1

Simplify each of the following.

- If
**,**then*x*^{2}= 25. -
*x*= 5 or*x*= –5 - Because
*x*could be either value, a rule is established. If a radical expression could have either a positive or a negative answer, then you always take the positive. This is called the “principal root.” Thus, - If
**,**then -
*x*^{3}= 64 and*x*= 4 - If
- If
- If
*x*^{2}= –4. There is no real value for*x*, so

Following are true statements regarding radical expressions.

n

a

Example

| even | positive negative zero | positive not real zero |

| odd | positive negative zero negative zero | positive negative zero not real zero |

When variables are involved, absolute value signs are sometimes needed.

Example 2

Simplify

It would seem that *x* is nonnegative. Because of this, *x*|, which guarantees that the result is nonnegative.

Absolute value signs are *never* used when the index is odd. Absolute value signs are sometimes used when the index is even, at those times when the result could possibly be negative.

Example 3

Simplify the following, using absolute value signs when needed.

- Since 2
*x*^{2}*y*^{6}could not be negative even if*x*or*y*were negative, absolute value signs are not needed. - Since the expression could be negative if
*y*were negative, a correct way to represent the answer is |2*x*^{2}*y*^{5}|. Because only*y*could have caused the answer to be negative, another way to represent the answer is 2*x*^{2}|*y|*^{5}. - Absolute value signs are never used when the index is odd.