 Properties of Radicals
 Cautions when Applying the Properties
 Applying the Properties of Radicals
1. Properties of Radicals
There are several properties of radicals we can apply to simplify expressions involving radicals. The following properties are generally true whenever n is greater than 1, and a and b are both positive real numbers:
and
Product Property:
Quotient Property:
Fractional Exponents:
2. Cautions when Apply the Properties
Avoid these common errors when applying properties of radicals:

 The properties of radicals only apply to factors; they do not apply to terms. For example, we can use the product property of radicals to break into two radicals because . However, we cannot break into .

 We can only bring an exponent outside of a radical if it applies to everything underneath the radical. For example, we can rewrite as because the exponent of 2 applied to everything underneath the radical. However, This is because the exponent of 2 applies only to the x, not the 16. (We could rewrite the expression as because 16 = )

 Taking the oddroot of a negative number leads to a real number solution, because a negative value raised to an odd exponent is negative. However, taking the evenroot of a negative value leads to a nonreal solution, because a negative value raised to an even exponent is never negative.
3. Applying the Properties of Radicals
When we recognize products, quotients, and powers with radicals, we can apply the properties of radicals to simplify the expression. This is shown in the examples below:






Product property






Our Solution







Quotient property



One factor of x cancels; one factor of y cancels



Product Property



Quotient property (and



Quotient property (and



Our Solution

We can use the properties of radicals to simplify expressions and solve equations. There are some cautions when applying the properties. The properties of radicals apply only to factors, numbers, and variables combined by multiplication, not by addition or subtraction.