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Recall that the power of a product rule is valid for rational exponents as well as integers. For example, the power of a product rule allows us to write

(4y)1/2 = 41/2 · y1/2 and (8 · 7)1/3 = 81/3 · 71/3.

These equations can be written using radical notation as

The power of a product rule (for the power 1/n) can be stated using radical notation. In this form the rule is called the product rule for radicals.

The nth root of a product is equal to the product of the nth roots. In symbols,

provided that all of the expressions represent real numbers.

The numbers 1, 4, 9, 16, 25, 49, 64, and so on are called perfect squares because they are the squares of the positive integers. If the radicand of a square root has a perfect square (other than 1) as a factor, the product rule can be used to simplify the radical expression. For example, the radicand of has 25 as a factor, so we can use the product rule to factor into a product of two square roots:

When simplifying a cube root, we check the radicand for factors that are perfect cubes: 8, 27, 64, 125, and so on. In general, when simplifying an nth root, we look for a perfect nth power as a factor of the radicand.

Example 1

Using the product rule to simplify radicals

Simplify each expression. Assume all variables represent positive numbers.

Solution

a) The radicand 4y has the perfect square 4 as a factor. So

b) The radicand 18 has a factor of 9. So

c) The radicand 56 in this cube root has the perfect cube 8 as a factor. So

d) The radicand in this fourth root has the perfect fourth power 16 as a factor. So