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 = 16·3,
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
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as the positive or principal nth root of a for
appropriate values of a and n. An alternative notation for 
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 
.
, 
also can be written using radicals. 
are real numbers:
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. 
