Rational Exponents

Definitions
Rational exponents are defined in terms of roots.
They are defined so that all the rules in the previous section for working
with integer exponents also work for the case of rational exponents.

For any real number (except that cannot be negative when the root is even),
and natural numbers,
is defined as the principal root of .

When is an even number and is positive, the principal root is always a positive number.
For example,
     .

When is a negative number, an even root of is not a real number.
Look at , which is the square root of   .
Suppose it has an answer     within the real number system.
Then we want to find a real number such that ,
by the definition of square root.
If the square root exists in the real number system, the root must be either
a negative number, zero, or a positive number. But the square of a negative
number or a positive number is positive, and the square of zero is zero, so
there is no number within the real number system that works.
Square roots, in general even roots, of negative numbers are called complex,
or imaginary, numbers.
Later in the course we will learn much more about complex numbers.

When is an odd number, can be positive or negative, and the root is a
real number.
For example,
     

In general,

                   
                

Note that    is not a real number because   
is not a real number.

See Examples 1 - 4, pages 53 - 56.

For the definition of rational exponents in terms of radicals, see the next section, Radicals.


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