Overview
Exponents are expressed as numbers or variables in superscript above a base number. When the exponent is equal to or greater than 2, it directs how many times the base number is to be multiplied. The numbers 1 and 0 are special exponents, and negative integers as exponents stand for reciprocal values.
Definition
Exponential notation has two parts, a base b and an exponent n. It is usually written as bn, with the superscript to the right of the base, and sometimes as b^n, which essentially means to raise n to the exponent position. The meaning of exponential notation depends on the value of the exponent, whether it is 2 or greater, whether it is 1 or 0, or whether it is negative.
Figure 1: An example of a base and exponent. 23 = 2∙2∙2 = 8.
Exponents of Two or Greater
If the value of the exponent n is 2 or greater than 2, it is a direction to multiply the base b n times. Suppose the number is 37. That means 3∙3∙3∙3∙3∙3∙3, which equals 2187. If the number is n2, that means n∙n, or “n squared”. The area of a square is also n∙n. If the number is n3, that means n∙n∙n, or “n cubed”. The volume of a cube is also n∙n∙n.
Exponents of One or Zero
Exponents of 1 or 0 follow a pattern. Any number n1, when n is greater than 0, is just the number itself, or n. Think of it as not multiplied at all. Therefore 31 equals 3, 451 equals 45, and 370,906,2321 equals 370,906,232. Any number n with an exponent of 0, by convention, equals 1. The number 70 equals 1, the number 2700 equals 1, and the number 468,213,9960 equals 1.
Figure 2: The definition of zero exponents in symbol form.
Negative Exponents
Negative integers as exponents also follow a pattern. Any exponent b-n, when b is not 0 means the same thing as 1/bn. For example, 3-2 means the same thing as 1/32, or 1/9. If the base is 10, the exponents in decimal notation follow a familiar pattern, so that 100 equals 1, 101 equals 10, 102 equals 100, 103 equals 1000, and so on. Also, 10-1, equals 1/10, or 0.1, 10-2 equals 1/100, or 0.01, 10-3 equals 1/1000 or 0.001, and so on.
Figure 3: The definition of negative exponents in symbol form.
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