**Radical Notations:** If is a real number, is a positive integer, and is a real number, then

**Rules for Radicals:** For all real numbers and , and positive integers and for which the indicated roots are real numbers:

1. Product Rule:

2. Quotient Rule:
,

3. Power Rule:

**Rationalizing Denominators:** No denominator contain a radical.

Exs.
,

**Simplified Radicals:** An expression with radicals is simplified when all of the following conditions are satisfied.

1. The radicand has no factor raised to a power greater than or equal to the index.

Ex.
.

2. The radicand has no fractions.

Exs.
,
.

3. The denominator is rationalized.

Exs.
,
.

4. Exponents in the Radicand and the index of the radical have no common factor.

Ex.
.

5. All indicated operations have been performed (if possible).

Ex.
.

**Like Radicals:** Radicals with the same radicand and the same index.

Ex.
and
are like radicals.

Department of Mathematics