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We have defined as the positive or principal nth root of a for appropriate values of a and n. An alternative notation for uses radicals.

RADICALS

If n is an even natural number and a > 0 or n is an odd natural number, then

The symbol is a radical sign, the number a is the radicand, and n is the index of the radical. The familiar symbol is used instead of .

EXAMPLE 1

Radical Calculations

With written as , also can be written using radicals.

The following properties of radicals depend on the definitions and properties of exponents.

PROPERTIES OF RADICALS

For all real numbers a and b and natural numbers m and n such that and are real numbers:

Property 3 can be used to simplify certain radicals. For example, since 48 = 16ยท3,

To some extent, simplification is in the eye of the beholder, and might be considered as simple as . In this textbook, we will consider an expression to be simpler when we have removed as many factors as possible from under the radical.

EXAMPLE 2

Radical Calculations