The Multiplication Property of Radicals allows us to write a single radical as the product of two radicals.

 Here are two examples: Square Root Cube Root

The property can also be used to write the product of two radicals as a single radical, provided the two radicals have the same index.

 Here are two examples: Square Root Cube Root

Property â€” Multiplication Property of Radicals

English The nth root of a product is the product of the nth roots.

Algebra where and are real numbers and n is a positive integer.

Example

Note:

If x is negative, then

For example, if x = -3:

In the remainder of this Topic, we will assume that each variable under a radical represents a nonnegative real number.

Example

Simplify:

a.

b.

Solution

To simplify, use the Multiplication Property of Radicals to rewrite each radical.

 a. Write the radical as a product of radicals. Simplify each radical. = 5 Â· x = 5x b. Write the radical as a product of radicals. Simplify

Note:

If possible, write the radicand as a product where one exponent is a multiple of the index. Use the largest multiple possible.

Here are three examples: