BMCC: Math Tutorials: Numbers:
Exponential Notation
Quick links to topics on this page:
Exponents and Exponential Notation
Some Definitions and Terms
Simplification of Expressions Containing Exponents
Converting Longhand Expressions to Exponential Notation
Powers of Ten
In the last two sections there were several instances where we wanted to represent a number multiplied by itself two or more times. Each time we wrote out the equation in "longhand," such as 4 x 4, or 2 x 2 x 2. Exponential notation is a "shorthand" method for representing repeated multiplication of a factor, in which you only need to write down the factor once, and indicate with another number how many times the factor should be multiplied by itself.
To indicate the factor 4 multiplied by itself 3 times, we can either write out the expression as we have been, like this:
or we can write the same expression in exponential notation, like this:
The number written as a superscript, in this case 3, is the power to which the first number, in this case 4, is raised. The superscript number is also known as the exponent. To read the left-hand side of the above expression out loud, one would say
In either case, the way that 43 is interpreted is 4 multiplied by itself 3 times, so
Another example:
and another:
Any number n raised to the first power (n1), such as 41, is equal to the number n itself. Thus
Generally, the exponent 1 is not written-- instead of writing 41 in an equation, you would just write 4.
And, although it may not be intuitively obvious, by definition any number raised to the zeroeth power equals 1 (n0 = 1).
When an exponent is either 2 or 3, there are specific terms that are sometimes used. A number n raised to the second power (n2) is also said to be squared ("n squared"), as in
A number n raised to the third power (n3) is also said to be cubed ("n cubed"), as in
To simplify an expression containing exponents, write the exponential term out in longhand and then perform the indicated calculations. To simplify 53 we would convert it to the equivalent longhand expression 5 x 5 x 5, and then multiply the terms to yield the product of 125.
Another example . . .
The process is the same for expressions that contain more than one exponential term.
and another example . . .
If an expression is written in longhand, it can be converted to exponential notation if any factors are multiplied more than once. The exponent for a factor is equal to how many times that factor appears in the formula. In the longhand expression 2 x 2 x 4 x 4 x 4 x 6, the factors 2 and 4 both appear more than once and can be expressed in exponential notation. The factor 2 appears twice, so it can be represented as 22. The factor 4 appears three times, so it can be expressed as 43. The factor 6 appears only once-- it could be expressed as 61, but recall that the exponent 1 is usually omitted, so the number 6 will remain unchanged. Thus,
Note that as long as all of the terms are being multiplied, the order of appearance does not matter. You can still count how many times a factor appears, and the exponent for that factor will be equal to the number of appearances. In the expression
the factors 2, 3, and 4 all appear more than once. They can be collected together and expressed as exponents. The factor 2 appears twice, 3 appears three times, and 4 appears twice. Therefore,
Most of our daily experience with numbers uses the base ten number system. The relation between common numerical terms and their respective expressions as exponents is shown below:
| Ten | 10 | 10 | 101 |
| Hundred | 100 | 10 x 10 | 102 |
| Thousand | 1,000 | 10 x 10 x 10 | 103 |
| Ten Thousand | 10,000 | 10 x 10 x 10 x 10 | 104 |
| Hundred Thousand | 100,000 | 10 x 10 x 10 x 10 x 10 | 105 |
| Million | 1,000,000 | 10 x 10 x 10 x 10 x 10 x 10 | 106 |
Next: Some Rules of Operation
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