 Exponents: Definition

In this lesson, the student is introduced to exponents. When a number is used repeatedly as a factor in multiplication, the number can be written as an exponent. The base of an exponent is defined as the number that is multiplied repeatedly. The exponent is defined as the number of times it is used as a factor.

The written format and terminology for reading and writing exponents is illustrated. Numbers written with exponents are referred to as exponential numbers. The size of the exponent is defined as the power to which the base is raised.

Any number raised to a power of one is the number itself.

Any number raised to a power of zero is 1.

There are two types of problems in this lesson: 1) students are given a repeated multiplication statement and must write it in exponent notation, and 2) students are given a number in exponent notation and must give the value.

Only positive base numbers are used in this lesson.

Negative Numbers Raised to a Power

When the base number in a number written in exponent format is negative, it means the negative number is multiplied repeatedly. The negative number is used as a factor for a number of times equal to the exponent.

When a negative number is raised to a power of one, the result is the negative number itself. When a negative number is raised to a power of zero, the result is 1. Zero raised to the power of zero is undefined.

When a negative number is raised to an even number power, the result will be a positive number. When a negative number is raised to an odd number power, the result will be a negative number. Each of these is illustrated with examples.

An important point about exponent notation with negative numbers is also illustrated. The proper notation for raising a negative number to a power is to enclose the negative number in parentheses and place the exponent outside the parentheses. When this is done, the negative number is used as the base and multiplied repeatedly.

When the negative number is raised to a power without parentheses, this represents a different situation. When this is the case, the positive number becomes the base that is used in repeated multiplication. The negative sign is then associated with the result, so raising a negative number to a power without parentheses always results in a negative number. The appropriate terminology for each of these situations is presented.

There are two types of problems in this lesson: 1) students are given a repeated multiplication statement with negative numbers as factors and must write it in exponent notation, and 2) students are given a negative number raised to a power in exponent notation and must give the value of the number. In the second type, some of the negative numbers are written in parentheses and some are not.

Only negative numbers are used as bases in this lesson.

Order of Operations with Exponents

In earlier lessons, students have learned the order in which operations are to be performed in evaluating expressions that use the four basic operations: addition, subtraction, multiplication, and division.

In this lesson, students learn that if an expression includes an exponential number, that number must be evaluated before any of the other operations are carried out.

Examples are given of how the order of operations are to be performed when the expression contains a term with an exponent. Students are given an expression such as these and are required to evaluate them.

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