Multiplying and Simplifying Radicals The Product Rule for Radicals is given by: Note that both of the radicals on the left have the same index. Throughout this section and the rest of the chapter we will assume that variables are nonnegative, and thus not need the absolute value sign.

Example 1 Multiply the radicals

The Product Rule for Radicals can be used in another way The Product Rule for Radicals can be used in another way. Rather than using it for multiplication, it can also be used to break up a radical expression. This will help in simplifying radicals.

Example 2 Simplify: Since the radical is a square root, we want to factor the radicand where one of the factors is a perfect square. Now apply the rule.

Example 3 Simplify: Since the radical is a square root, we want to factor the radicand where one of the factors is a perfect square.

Now apply the rule.

When factoring a radicand that includes powers of variables, try to get a variable factor with an exponent that is a multiple of the index n. Example 4 Simplify: We have an index of 5, so we want exponents that are multiples of 5.

Now apply the rule.

In this last problem we used … In the future, we will skip the middle step, and use the shortcut of dividing the exponent by the index, leaving out the middle step.

Example 5 Simplify: Since the radical is a cube root, we want to factor the radicand where one of the factors is a perfect cube.

Example 6 Multiply and simplify: Multiply using the product rule.

Since the radical is a fourth root, we want to factor the radicand where one of the factors is a perfect fourth power.

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