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Exponents

Three to the fourth power with the exponent of four written as a small number above and to the right of the base number three.

An exponent tells how many times to multiply a number (the ‘base’ number).

34 = (3)(3)(3)(3) = 81

xn = (x)(x)(x)

(x is mulitpled n times)

Exponents are shorthand: it is much quicker to write 54 than (5)(5)(5)(5)

Examples:

32 = (3)(3) = 9
33 = (3)(3)(3) = 27
34 = (3)(3)(3)(3) = 81

Example: Solve 54

  1. Multiply the base by itself: (5)(5) = 25
  2. Multiply the product times the base (third time): (25)(5) = 125
  3. Multiply the product times the base (fourth time): (125)(5) =625

52 = (5)(5) = 25
53 = (5)(5)(5) = (25)(5) = 125
54 = (5)(5)(5)(5) = (125)(5) = 625
55 = (5)(5)(5)(5)(5) = (625)(5) = 3125

Pattern of Positive and Negative Exponents

A positive base raised to a positive number gets larger as the exponent gets higher.
An exponential term with a negative exponent gets smaller as the exponent gets further below zero.

Example: Pattern of Positive and Negative Exponents with Base 4
44 (4·4·4·4) 256  
43 (4·4·4) 64
42 (4·4) 16
41 4 4
40 1 1
4-1 141 14 14
4-2 142 1(4·4) 116
4-3 143 1(4·4·4) 164
4-4 144 1(4·4·4·4) 1256

Notes:
x0 = 1
(except if x is a 0. 00 has no answer.)

1xa is the same as x-a. A term with a negative exponent is equal to the reciprocal of the term.

How are exponents useful?

Exponents are shorthand: it is much quicker to write 54 than (5)(5)(5)(5).

Exponents can represent change at a constant rate, for example:

  • A population doubling every year (e.g. population change rate = (p)(2x) where x = number of years)

Vocabulary

Base
A base is a number or term that is raised to a power, or multiplied by itself. In 6x3, x is the base. In (6x)3, 6x is the base.
Exponent
An exponent tells how many times to multiply a number or term (the "base" ). The exponent is called a power. In 63, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6).
Power
An exponent. A power tells how many times to multiply a number or term. In 63, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6).
Product
The number or expression that is the result of multiplying numbers and/or expressions.
Reciprocal
The reciprocal of a number is 1 divided by that number. The reciprocal of x is 1x. This can also be written as x-1 When you multiply a number by its reciprocal, the result is 1. e.g. (x)(1x) = 1.
Term
A term is a number, variable, or the product of a number and variable(s). These are each terms: 4, x, 4x, 4x2. In an expression, terms are separated by addition or subtraction operators (+,-).

 

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This material is based upon work supported by the National Science Foundation (NSF) under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.