4 Multiply square root radicals. For nonnegative real numbers a and b,andThat is, the product of two square roots is the square root ofthe product, and the square root of a product is the product ofthe square roots.It is important to note that the radicands not be negative numbers in the product rule. Also, in general,
5 Simplify radicals using the product rule. A square root radical is simplified when no perfectsquare factor remains under the radical sign.This can be accomplished by using the product rule:
6 EXAMPLE 2 Simplify each radical. Using the Product Rule to Simplify RadicalsSimplify each radical.
7 EXAMPLE 3 Find each product and simplify. Multiplying and Simplifying RadicalsFind each product and simplify.
8 Simplify radicals by using the quotient rule. The quotient rule for radicals is similar to the productrule.
9 EXAMPLE 4 Simplify each radical. Using the Quotient Rule to Simply RadicalsSimplify each radical.
10 EXAMPLE 5Using the Quotient Rule to Divide RadicalsSimplify.
11 EXAMPLE 6Using Both the Product and Quotient RulesSimplify.
12 Simplify radicals involving variables. Radicals can also involve variables.The square root of a squared number is always nonnegative. The absolute value is used to express this.The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers
13 EXAMPLE 7Simplifying Radicals Involving VariablesSimplify each radical. Assume that all variables represent positive real numbers.Solution:
14 Simplify other roots.To simplify cube roots, look for factors that are perfectcubes. A perfect cube is a number with a rational cube root.For example, , and because 4 is a rationalnumber, 64 is a perfect cube.For all real number for which the indicated roots exist,