Powers and Exponents
Powers and Exponents
An exponent is a number that indicates how many times a certain number, say b, is multiplied by itself. The expression 2 × 2 × 2 × 2 can be written as 2^{4}, where 4 is the exponent and 2 is called the base. An exponent, which is also called a power, is written as a superscript of the base number. A base b raised to a power n is written as b ^{n}.
Exponent is a simple but powerful idea and can be used to create shortcuts in problems. To multiply 2^{4} by 2^{6}, for instance, simply add the powers to find the product 2^{4 + 6}, or 2^{10}.
What if 2^{4} is multiplied by 3^{6}? The bases, 2 and 3, are different. In this case, the product cannot be found by adding the powers. The following are the three basic rules of exponents. Using these three laws, more properties of exponents can be found.
 When multiplying two numbers with the same base, add the exponents: b ^{n} × b ^{m} = b ^{n + m }
Example: 3^{3} × 3^{8} = 3^{3 + 8} = 3^{11}  When dividing two numbers with the same base, subtract the exponents:
Example:  When raising a power to a power, multiply the exponents: (b ^{n})^{m} = b ^{n × m}
Example: (4^{3})^{2} = 4^{3 × 2} = 4^{6}.
Zero Exponent
An interesting rule involving exponents is that a number raised to zero power, say b ^{0}, is equal to 1. This surprising result follows directly from Rule 2. Recall, a number divided by itself is 1.
Apply Rule 2.
Therefore, b ^{0} = 1. This means that any number raised to 0 is 1. Hence, 5^{0}, 2^{0}, and 31^{0} are all equal to 1.
Negative Exponent
What does a negative exponent mean? Here is another rule that also follows from Rule 2.
Using b ^{0} = 1 and b ^{1} = b, can be expressed as follows.
Apply Rule 2 to the righthand side.
Therefore,
So,
Fractional Exponent
A base raised to a fractional power, say b ^{1/2}, is another way to express the square root of b.
Therefore, . Similarly, b ^{1/3} is another way to express the cube root of b.
Therefore, . In a general case, the n th root of b is b ^{1/n}.
Combining Rule 1 and the fractional exponent rule results in the following exponent property.
For instance, . The number within the parenthesis, square root of 4, is 2.
4^{5/2} = (2)^{5}
4^{5/2} = 2 × 2 × 2 × 2 × 2
4^{5/2} = 32
see also Radical Sign.
Rafiq Ladhani
Bibliography
Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.
Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: AddisonWesley, 2001.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Powers and Exponents." Mathematics. . Encyclopedia.com. 21 Mar. 2019 <https://www.encyclopedia.com>.
"Powers and Exponents." Mathematics. . Encyclopedia.com. (March 21, 2019). https://www.encyclopedia.com/education/newswireswhitepapersandbooks/powersandexponents
"Powers and Exponents." Mathematics. . Retrieved March 21, 2019 from Encyclopedia.com: https://www.encyclopedia.com/education/newswireswhitepapersandbooks/powersandexponents
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.