Exponents and Roots

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Integer Exponents

Recall that , while , and so on. In this section a more general meaning is given to the symbol .

DEFINITION OF EXPONENT

If n is a natural number, then

where a appears as a factor n times.

In the expression n is the exponent and a is the base. This definition can be extended by defining for zero and negative integer values of n .

ZERO AND NEGATIVE EXPONENTS

If a is any nonzero real number, and if n is a positive integer, then

(The symbol is meaningless.)

EXAMPLE 1

Exponents

The following properties follow from the definitions of exponents given above.

PROPERTIES OF EXPONENTS

For any integers m and n, and any real numbers a and b for which thefollowing exist:

EXAMPLE 2

Simplifying Exponential Expressions

Use the properties of exponents to simplify each of the following. Leave answers with positive exponents. Assume that all variables represent positive real numbers.

CAUTION

If Example 2(e) were written , the properties of exponents would not apply. When no parentheses are used, the exponent refers only tothe factor closest to it. Also notice in Examples 2(c), 2(g), 2(h), and 2(i) that anegative exponent does not indicate a negative number.