Chapter 1 Elementary Arithmetic Section 1.4 Powers and Roots
1.4.1 Exponentiation and RootsThe following section deals with expressions of the form , where . But the question is: For which numbers can this power be reasonably defined?
Powers with positive integer exponents are defined as follows:
Here, some special cases exist that you should ideally know by heart:
For a zero exponent, the value of the power is one, i.e. for example, , also . But for a zero base, for , we have . For base , we have
Powers with negative integer exponents are defined by the formula
Calculate the values of the following powers.
However, even for a rational exponent of the form , we need to extend this definition again so that we can calculate , for example. This power can be expressed as a root as well, namely . Generally, we have:
Let and with The -th root has the power representation .
For with we have the following calculation rules:
- Two roots with the same exponent are multiplied by multiplying the radicands and extracting the root of the product, leaving the exponent of the root unchanged:
- Two roots with the same exponent are divided by dividing the radicands and extracting the root of the quotient, leaving the exponent of the root unchanged:
A general power with rational exponent is then calculated as follows:
The calculation rules for powers with real base and rational exponent are known as exponent rules. The rules vary depending on whether powers of the same base or the same exponent are considered.
Calculate the following roots (here, the result is ):