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Applying the Properties of Radicals

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This lesson applies several properties of radicals in order to simplify expressions.

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- Properties of Radicals
- Cautions when Applying the Properties
- Applying the Properties of Radicals

**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:

**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 5 • 3 = 15. 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 = 4^{2})

Taking the odd-root of a negative number leads to a real number solution, because a negative value raised to an odd exponent is negative. However, taking the even-root of a negative value leads to a non-real solution, because a negative value raised to an even exponent is never negative.

**Applying the Properties of Exponents**

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:

EXAMPLE 1

EXAMPLE 2