# What is the value of 1 to the power of 1

## Powers and rational numbers

bettermarks »Math book» Numbers »Rational numbers» Arithmetic, arithmetic laws and word problems with rational numbers »Powers and rational numbers

In these explanations you will learn how to calculate with powers of rational numbers.

### Basic concepts for potencies

Each power consists of an exponent and a base. You pronounce the arithmetic operation as “2 to the power of 5”. If there is a “2” in the exponent, for example at, then you can also say “7 squared”.
,,, ... are called powers of ten.
,,,, ... are called powers of two.

### Convert potencies into a product

The power notation is an abbreviation for multiplying equal numbers. The natural number in the exponent indicates how often the base is multiplied by itself. One also uses powers with the exponents 1 and 0. A power with the exponent 1 represents the number itself, i.e. the base: A power with the exponent 0 represents the number 1 for every base (except zero):; ; ; ...
A power is the repeated multiplication of a number by itself!
When the exponent is 1, the power is equal to the base.
If the exponent is zero and the base is nonzero, the power is 1.
Natural numbers as a basis
Negative numbers as a basis

### Powers with fractions

If the base of a power is a fraction, an easily recognizable calculation rule follows from the definition of powers: You can convert a power with a fraction as the base by dividing the exponent between the numerator and denominator.

### Signs of powers

In the case of powers, the following rules apply to the signs:
If the base is positive, the total potency is always positive.
If the base is negative, the entire power is positive for even exponents.
If the base is negative, the entire power is negative for odd exponents.
Negative base with an even exponent
Any two negative factors can be combined into one positive factor. The product of these positive factors is also positive.
Negative base with odd exponent
Any two negative factors can be combined into one positive factor. However, if the exponent is odd, the number of factors is odd. Thus you form the product of all positive factors and one negative factor and get a negative result.
Different bases and exponents in comparison
With a positive base (here the 2) the entire power is always positive, with a negative base (here the -2) the sign of the result always changes, depending on whether the exponent is odd (e.g. 1) or even (e.g. 2) is.