Pronunciation key ( ik-spō′nənt ) |
ex•po•nent
adj.
[L. exponens, exponent- ppr. of exponere to put forward; ex- out + ponere to put. See EXPOUND].
- Expository, explanatory; expounding; interpreting.
n.
- One that expounds or interprets principles, methods, etc.
- One that advocates, represents or speaks as, or a thing that is an example or symbol of something.
- Algebra. A number or symbol placed at the right and above another figure, expression or symbol to show how many times the latter is to be raised (multiplied by itself). For instance (b4 = b x b x b x b).
In mathematics an exponent is a symbol or number that designates how many times the factor or base, is raised, or multiplied by itself to form a product or repeated as a factor.
An exponent is always located to the top-right of another numeral called the base. Whole number exponents indicate the number of successive times a base is to be used as a multiplier. In the expression (d4) d is the base or factor and 4 is the exponent. Therefore the symbol, 4 is indicative that
d X d X d X d = d4
Further algebraic-exponential expressions,
53
The superscript 3, is the exponent. It is the equivalent of,
( 5 • 5 • 5 ) or ( 5 x 5 x 5 )
The exponential form can only be applied if factors constituting it, are identical.
The exponent is also "n to the power of".
Therefore, n2 can be stated as n raised to the second power (or n squared). Typically, in algebra, dx is stated as,
d to the xth, or the xth power of d.
Exponents can also be substituted for zero, negative numbers and fractions. Any nonzero quantity to the zero power equals 1.
b0 = 1 for b ≠ 0
A base or factor, with a negative exponent equals the reciprocal of the base but with the positive exponent, or the base with the positive exponent is divided into 1.
For instance,
n-3 = 1/n3
A fractional exponent indicates the extraction of a root of a factor; such as,
n raised to the ½ power equals the square root of n.
Exponential calculations can be made using the following basic rules,
- Multiplication of exponential factors which have the same factor but different powers, the powers are added together. Such as,
- Raising an exponential factor to a power, is accomplished by multiplication of the exponents
- The division of exponential factors which have the same base but different powers, apply subtraction to the exponents, for example,
da dh = d a + h
( b x )d = bxd
bn / b x = b n - x
Laws of exponents provide a convenient and concise means of expressing large and small numbers and to make calculations. For example, the number 100,000 is equivalent to
10 X 10 X 10 X 10 X 10
when expressed in exponential terms is
105
Similarly, the small number .0001 is equal to
1/(10 X 10 X 10 X 10) or 10-4
The problem
may be expressed as,
106 X 10-4 X 10-6 = 106-4-6 = 10-4 = 1/104 = 1/10,000 = .0001
Many numbers have few significant purposes. One of them, 30,000,000 is the modulus of elasticity of steel. The number 3 is the only important number, and in .000006, the temperature coefficient of expansion for steel, only the 6 is important.
30,000,000 is equal to,
3 X 10 X 10 X 10 X 10 X 10 X 10 X 10 or 3 x 107
.000006 is equivalent to,
6 X 1/(1,000,000) = 6 X 1/106 = 6 X 10-6
Equations that contain three important digits, such as
326,000 are equal to 3.26 X 105, and .00497, equal to 4.97 X 10-3
are common in physical science applications.
The prior examples the value of the exponent is equal to the number of digits between the present location of the decimal point and its intended location. For numbers greater than 1 the exponent is positive (+) and for numbers less than 1, the exponent is negative (-). In the conversion from scientific notation, or powers of ten into ordinary numbers the same rules apply. Likewise, a positive exponent must result in a greater value and a negative exponent results in a reduced value.
An example is to convert
2.3 X 105
to ordinary notation, the decimal point is moved 5 spaces to the right or, 230,000. If the figure had contained a negative exponent,
2.3 X 10-5
it would have been changed to an ordinary notation by moving the decimal point 5 spaces to the left, that is, .000023.
It is not necessary that this scientific notation use a number between one and ten times a power of 10. For instance, 32,500,000 might be expressed as
32.5 X 106 though it may also be expressed as 3.25 X 107
At times a zero (0) is a number of importance and must be indicated. An example of this being the first four digits of the number would be expressed in scientific notation as 3.250 X 107.
Calculations are greatly simplified when the numbers are expressed using scientific notations. For example,
2,200,000 X .00003 = (2.2 X 106) X (3.0 X 10-5) = 6.6 X 10 = 66;
= 2.0 X 10-8 (-10) = 2.0 X 102 = 200.
Using exponents in multiplication and division is the foundation for logarithms. Logarithms are exponents to special bases, notably 10 and e = 2.7183 (e is constant which occurs frequently in science). Tables that provide the power of a particular number for any given exponent are available, such as 2 can be shown as the equivalent of 10.30103 and .30103 is the logarithm of 2 to the base 10 expressed as log 102 = .30103.
2 X 3 X 3 X 3 X 3
can be written in exponential form as
2 X 34
That is, 4 is the exponent on the base 3. The value 34 is read as "3 to the 4th" or the "4th power of 3." and is equal numerically to 3 X 3 X 3 X 3 = 81. From this definition it is possible to develop mathematical laws. See the following table.
Image credit: The American Peoples Encyclopedia ©1960
- Law one. A number raised to the first power is equal to the number itself.
- Law two. The exponents involved in multiplication are added.
- Law three. The exponents involved in division are subtracted.
- Law four. When a base and its exponent are raised to an additional exponent, the exponents involved are multiplied.
- Law five. A number raised to a fractional exponent is equal to the exponential root of the number. (This is actually a corollary of Law I.)
- Law six. A number raised to a fractional exponent whose numerator is not one, is equal to the root of the numerator and the power of the denominator. (This is actually a corollary of Law IV and V.)
- Law seven. A number raised to a negative exponent is equal to the reciprocal of the number raised to its positive exponent. (This is a corollary of Law III.)
- Law eight. A number raised to the zero power is equal to (1) one.
The first four laws apply to whole numbers and positive exponential integers, but can be applied to formulate further laws,
The laws of exponents are derived from the laws that govern positive integers and exponents, which lead to further meanings for those with fractional, negative, and zero exponents. An exponent is any expression that can be performed in accordance with the laws of exponential operations derived from those for positive integers.
Laws of exponents for whole numbers provide meaning to the expression
ax only if x is a rational number.
A rational number is that which can be expressed as an integer or quotient of integers. If
x
were an irrational number, such as
or 
it cannot be expressed ax into terms of powers and roots.
However, if the irrational number is approximated as closely possible in rational numbers then the approximation of
s in a8
will have a decimal value and the rules of exponents can be applied.
The function
y = 2x
when x is a real number, takes the form of the curve represented in the graph (below). This is known as an exponential curve. It touches the x-axis only at infinity and approaches the limit value of x.
Image credit: The American Peoples Encyclopedia ©1960
An Exponential Curve.
According to this curve when x is 2, y is 4; when x is 3, y is 8. If x is an irrational number such as or 1.414, the corresponding value of y could be approximated from the curve as y = 2.664 provided, of course, the curve were drawn accurately enough. Graphs of the function y = ax are all similar for all values of a greater than 1.
A base which is used frequently in exponential functions are an irrational number known as e and has a numeric value of 2.7182… it is defined in terms of limits as
e = lim (1+1/n)n
This formula serves numerous functions in science and business, as a mathematical expression of laws which deal with the rate of growth and decay.
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