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Multiplying the number 3 by itself four times gives
which can be written more simply with the shorthand notation
The notation 3 4 is read as “3 raised to the fourth power” or “3 to the fourth power.” In this example, the number 3 is referred to as the base, and the number 4 is called the exponent. In the natural sciences the most frequently used base is an irrational number with the symbol e , whose value to three decimal places is 2.718. An irrational number is a number that cannot be written as a fraction. e.g., the square root of 2 ( ) and the number pi (π).
The most general representation of exponentiation is
where a and m can be arbitrary numbers. A few numerical examples are
The first two examples can be verified by hand calculation; the last two are easy to verify on a calculator.
The rule for multiplying exponentials follows from the definition of exponentials:
This rule is easily verified by checking an example:
All the other properties of exponentials follow directly from the rule for combining exponents.
We use the previous rule to deduce what a to the 0th power ( a 0 ) means. To make the point concrete, take a = 3. Equation AB.1 allows us to write
This expression is true if, and only if, 3 0 = 1. In general, any non-0 number raised to the 0th power is equal to 1:
The reason for excluding 0 from the definition will be clear shortly.
To deduce the meaning of a negative exponent, again take a = 3 as the base and use Equation AB.1 to write
This expression is true if, and only if, 3 −2 = 1/3 2 . In general,
The definition in Equation AB.3 extends the rule for combining exponents ( Equation AB.1 ) to include division of exponentials:
We now see why in the definition of the 0th power ( Equation AB.2 ), the base a cannot be 0: a = 0 in Equation AB.4 would force a division of 0 by 0—an operation that has no meaning.
Combining exponents shows what happens when an exponential is raised to another power, for example:
In general,
We now investigate the case in which the exponent is a fraction (i.e., a m / n ). Taking a concrete example with a = 7, we ask what is meant by 7 1/2 . Applying Equation AB.1 gives
which immediately shows that 7 1/2 = . In general:
A direct consequence of combining Equations AB.6 and AB.4 is that
Because any rational decimal number can always be written as a fraction (e.g., 1.5 = 3/2, 0.47 = 47/100), decimal numbers in the exponent can be dealt with as fractions.
Equations AB.1 to AB.7 constitute the properties of exponentials that are important for computation. These properties are summarized in Box AppB.1 .
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