# The Product Types Ontology: Class Definition for "Exponentiation"

This page is part of http://www.productontology.org/, a huge, precise dictionary of product types and brand names for marking up Web sites with schema.org or the GoodRelations e-commerce standard.

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### pto:Exponentiation(rdf:type owl:Class)

URI http://www.productontology.org/id/Exponentiation
rdfs:subClassOf gr:ProductOrService, http://schema.org/Product
rdfs:label Exponentiation (as a class or brand name of products of services)
Translation(s):  Magsverheffing@af;  Potenz (Mathematik)@als;  ንሴት@am;  رفع (رياضيات)@ar;  Potenciación@ast;  Qüvvətə yüksəltmə@az;  Дәрәжәгә күтәреү@ba;  Ступеняванне@be;  Степенуване (математика)@bg;

rdfs:comment

Exponentiation is a mathematical operation, written as involving two numbers, the base and the exponent or power and pronounced as "(raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: <math display="block">b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}. The exponent is usually shown as a superscript to the right of the base. In that case, is called "b raised to the nth power", "b (raised) to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: <math display="block"> \begin{align} b^{n+m} & = \underbrace{b \times \dots \times b}_{n+m \text{ times}} \\ & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}} \\ & = b^n \times b^m \end{align} In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that b^0 must be equal to 1, as follows. For any n, b^0 \cdot b^n = b^{0+n} = b^n. Dividing both sides by b^n gives b^0 = b^n / b^n = 1. The fact that b^1 = b can similarly be derived from the same rule. For example, (b^1)^3 = b^1 \cdot b^1 \cdot b^1 = b^{1+1+1} = b^3 . Taking the cube root of both sides gives b^1 = b. The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what b^{-1} should mean. In order to respect the "exponents add" rule, it must be the case that b^{-1} \cdot b^1 = b^{-1+1} = b^0 = 1 . Dividing both sides by b^{1} gives b^{-1} = 1 / b^1, which can be more simply written as b^{-1} = 1 / b, using the result from above that b^1 = b. By a similar argument, b^{-n} = 1 / b^n. The properties of fractional exponents also follow from the same rule. For example, suppose we consider \sqrt{b} and ask if there is some suitable exponent, which we may call r, such that b^r = \sqrt{b}. From the definition of the square root, we have that \sqrt{b} \cdot \sqrt{b} = b . Therefore, the exponent r must be such that b^r \cdot b^r = b . Using the fact that multiplying makes exponents add gives b^{r+r} = b . The b on the right-hand side can also be written as b^1 , giving b^{r+r} = b^1 . Equating the exponents on both sides, we have r+r = 1 . Therefore, r = \frac{1}{2} , so \sqrt{b} = b^{1/2} . The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices. Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography. (Source: Wikipedia, the free encyclopedia, see http://en.wikipedia.org/wiki/Exponentiation)

Note: The extraction of the abstract from the Wikipedia page may sometimes yield imperfect results. We are improving the algorithm regularly.

Note: This is a generic list. Some of the properties may not be applicable to this particular type of object.

Trademark Disclaimer:  Since this service returns class descriptions for potentiall any series of characters, it cannot indicate automatically whether a name is a registered trademark or otherwise protected. We assume no liability for the absence of trademark rights and other damages. See the section "License" below for details.

## Usage

</div>
</div>
</body>
</html>


### Turtle Syntax

@prefix foaf: <http://xmlns.com/foaf/0.1/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .
@prefix pto: <http://www.productontology.org/id/> .
@prefix gr: <http://purl.org/goodrelations/v1#> .
@prefix foo: <http://example.com/> .

# The object
foo:myObject a <http://www.productontology.org/id/Exponentiation> ;
a gr:SomeItems ;
gr:name "... a short name for the object ..."@en ;
gr:description "... a longer description ..."@en .

# The agent (person or company) who is offering it
gr:legalName "ACME Corp" ;
gr:offers foo:Offer .

# The offer to sell it
foo:Offer a gr:Offering ;
gr:includes foo:myObject;
foaf:page <http://URI_of_the_page_containing_the_offer.com>;
gr:validFrom "2011-01-24T00:00:00+01:00"^^xsd:dateTime ;
gr:validThrough "2011-12-24T00:00:00+01:00"^^xsd:dateTime ;
gr:hasPriceSpecification
[ a gr:UnitPriceSpecification ;
gr:hasCurrency "USD"^^xsd:string ;
gr:hasCurrencyValue "19.99"^^xsd:float ;
gr:validThrough "2011-12-24T00:00:00+01:00"^^xsd:dateTime ] .


Note: Replace gr:SomeItems (http://purl.org/goodrelations/v1#SomeItems) by gr:Individual if you are describing a unique object of that kind (e.g. antique furniture).

### RDFa

<!DOCTYPE html>
<html version="HTML+RDFa 1.1" xmlns="http://www.w3.org/1999/xhtml">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<title>An offer to sell a / some Exponentiation</title>
<body>
<div xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#"
xmlns:foaf="http://xmlns.com/foaf/0.1/"
xmlns:gr="http://purl.org/goodrelations/v1#"
xmlns:pto="http://www.productontology.org/id/"
xmlns:foo="http://example.com/"
xmlns:xsd="http://www.w3.org/2001/XMLSchema#">

<!-- The agent (person or company) who is offering it -->
<div property="gr:legalName">ACME Corp</div>
<div rel="gr:offers">
<!-- The offer to sell it -->
<div rel="gr:includes">
<!-- The object -->
<div rel="rdf:type" resource="http://purl.org/goodrelations/v1#SomeItems"></div>
<div property="gr:description" xml:lang="en">... a longer description ...</div>
<div property="gr:name" xml:lang="en">.. a short name for the object ...</div>
</div>
</div>
<div rel="foaf:page" resource="http://URI_of_the_page_containing_the_offer"></div>
<div rel="gr:hasPriceSpecification">
<div typeof="gr:UnitPriceSpecification">
<div property="gr:hasCurrency" content="USD" datatype="xsd:string">\$ </div>
<div property="gr:hasCurrencyValue" datatype="xsd:float">19.99</div>
<div property="gr:validThrough" content="2011-12-24T00:00:00+01:00"
datatype="xsd:dateTime"></div>
</div>
</div>
<div property="gr:validFrom" content="2011-01-24T00:00:00+01:00"
datatype="xsd:dateTime"></div>
<div property="gr:validThrough" content="2011-12-24T00:00:00+01:00"
datatype="xsd:dateTime"></div>
</div>
</div>
</div>
</div>
</body>
</html>


Note: Replace gr:SomeItems (http://purl.org/goodrelations/v1#SomeItems) by gr:Individual if you are describing a unique object of that kind (e.g. antique furniture).

### RDF/XML

<?xml version="1.0" encoding="UTF-8" ?>
<rdf:RDF xmlns:gr="http://purl.org/goodrelations/v1#"
xmlns:pto="http://www.productontology.org/id/"
xmlns:xsd="http://www.w3.org/2001/XMLSchema#"
xmlns:foaf="http://xmlns.com/foaf/0.1/"
xmlns:foo="http://example.com/"
xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<!-- The object -->
<rdf:type rdf:resource="http://www.productontology.org/id/Exponentiation"/>
<rdf:type rdf:resource="http://purl.org/goodrelations/v1#SomeItems"/>
<gr:name xml:lang="en">... a short name for the object ...</gr:name>
<gr:description xml:lang="en">... a longer description ...</gr:description>
</rdf:Description>
<!-- The agent (person or company) who is offering it -->
<gr:legalName>ACME Corp</gr:legalName>
<gr:offers rdf:resource="http://example.com/Offer" />
<!-- The offer to sell it -->
<gr:includes rdf:resource="http://example.com/myObject" />
<foaf:page rdf:resource="http://URI_of_the_page_containing_the_offer"/>
<gr:validFrom rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">
2011-01-24T00:00:00+01:00</gr:validFrom>
<gr:validThrough rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">
2011-12-24T00:00:00+01:00</gr:validThrough>
<gr:hasPriceSpecification>
<gr:UnitPriceSpecification>
<gr:hasCurrency rdf:datatype="http://www.w3.org/2001/XMLSchema#string">USD</gr:hasCurrency>
<gr:hasCurrencyValue rdf:datatype="http://www.w3.org/2001/XMLSchema#float">19.99</gr:hasCurrencyValue>
<gr:validThrough rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">
2011-12-24T00:00:00+01:00</gr:validThrough>
</gr:UnitPriceSpecification>
</gr:hasPriceSpecification>
</gr:Offering>

</rdf:RDF>


Note: Replace gr:SomeItems (http://purl.org/goodrelations/v1#SomeItems) by gr:Individual if you are describing a unique object of that kind (e.g. antique furniture).

### SPARQL Query

prefix foaf: <http://xmlns.com/foaf/0.1/>
prefix xsd: <http://www.w3.org/2001/XMLSchema#>
prefix pto: <http://www.productontology.org/id/>
prefix gr: <http://purl.org/goodrelations/v1#>
prefix foo: <http://example.com/>

# Find the cheapest offer for a Exponentiation

SELECT * WHERE{
?company gr:offers ?offer .
?offer a gr:Offering .
OPTIONAL {?offer rdfs:label ?label } .
OPTIONAL {?offer gr:name ?label } .
OPTIONAL {?offer rdfs:comment ?label } .
OPTIONAL {?offer gr:description ?label } .
?offer gr:hasPriceSpecification ?p .
?p a gr:UnitPriceSpecification .
?p gr:hasCurrency ?currency .
?p gr:hasCurrencyValue ?price .
?offer gr:includes ?product .
?product a <http://www.productontology.org/id/Exponentiation> .
}
ORDER BY (?price)
LIMIT 10


You can also use the class definitions from this site for better describing the type of your page or product for the Facebook Open Graph Protocol.

Simply define the namespace prefix pto: in the <html> element of your page

<html version="HTML+RDFa 1.1"
xmlns="http://www.w3.org/1999/xhtml"
xmlns:pto="http://www.productontology.org/id/"
>

and use the compact URI (CURIE) pto:Exponentiation in combination with og:type as follows:

	<meta property="og:type" content="pto:Exponentiation"/>


A complete example is here.

## Contact Information

Univ.-Prof. Dr. Martin Hepp

E-Business and Web Science Research Group
Chair of General Management and E-Business
Universität der Bundeswehr München
Werner-Heisenberg-Weg 39
D-85579 Neubiberg, Germany

Phone: +49 89 6004-4217
eMail: mhepp(at)computer.org (preferred mode of communication)
Web: http://www.heppnetz.de/

## Caching Policy

In order to minimize the load on the Wikipedia API, all requests are cached internally for 72 hours. This means that changes to the English Wikipedia will be available in this service within 72 hours or less if the same entry has been requested before. Classes not requested within the last six hours are always guaranteed to be in sync with the latest version in Wikipedia.

The RDF/XML dump file is updated every 12 hours only.

Trademark Disclaimer:  Since this service returns class descriptions for potentially any series of characters, it cannot indicate automatically whether a name is a registered trademark or otherwise protected. If you want us to block a certain name, please send your request including proof of your rights on the name to our contact address listed below.. Any of the trademarks, service marks, collective marks, design rights or similar rights that are mentioned, used, or cited in this service are the property of their respective owners. Their use here does not imply that you may use them for any purpose other than for the same or a similar informational use as contemplated by the original authors of the underlying Wikipedia articles under the CC-BY-SA and GFDL licensing schemes. Productontology.org is neither endorsed by nor affiliated with any of the holders of any such rights and as such cannot grant any rights to use any otherwise protected materials. Your use of any such or similar incorporeal property is at your own risk.

## Acknowledgements

Thanks to Stefano Bertolo, Julien Chaumond, Bob Ferris, Kingsley Idehen, Axel Polleres, Andreas Radinger, Alex Stolz, and Giovanni Tummarello for very valuable feedback, and to Katharina Siorpaes and Daniel Bachlechner, who contributed to the initial analysis of the stability of Wikipedia URIs back in 2007.

The work on The Product Types Ontology has been supported by the German Federal Ministry of Research (BMBF) by a grant under the KMU Innovativ program as part of the Intelligent Match project (FKZ 01IS10022B).

## References

Wikipedia: Exponentiation, available at http://en.wikipedia.org/wiki/Exponentiation.

Hepp, Martin: GoodRelations: An Ontology for Describing Products and Services Offers on the Web, Proceedings of the 16th International Conference on Knowledge Engineering and Knowledge Management (EKAW2008), Acitrezza, Italy, September 29 - October 3, 2008, Springer LNCS, Vol 5268, pp. 332-347.

Hepp, Martin; Siorpaes, Katharina; Bachlechner, Daniel: Harvesting Wiki Consensus: Using Wikipedia Entries as Vocabulary for Knowledge Management, IEEE Internet Computing, Vol. 11, No. 5, pp. 54-65, Sept-Oct 2007.