The raising of a number to a natural power means its direct repetition by its own factor a natural number of times. A number repeated as a factor is the basis of the degree, and a number indicating the number of identical factors is called an exponent. The result of the actions taken is the degree. For example, three to the sixth power means repeating the number three as a factor six times.
The basis of the degree can be any number other than zero.
The second and third degrees of the number have special names. This, respectively, is a square and a cube.
For the first power of a number, this number itself is taken.
For positive numbers, a degree with a rational indicator is also defined. As everyone knows, any rational number is written in the form of a fraction whose numerator is an integer, but the denominator is natural, that is, a positive integer other than unity.
A degree with a rational indicator is the root of the degree equal to the denominator of the degree indicator, and the radical expression is the basis of the degree raised to a degree equal to the numerator. For example: three in 4/5 is equal to the root of the fifth degree out of three in the fourth.
We note some properties that follow directly from the definition in question:
- any positive number in a rational degree is positive;
- the value of a degree with a rational indicator does not depend on the form of its record;
- if the basis is negative, then the rational degree of this number is not defined.
With a positive basis, the properties of the degree are true regardless of the indicator.
Degree properties with a natural indicator:
1. Multiplying the degrees having the same base, leave the base unchanged and add up the indicators. For example: if you multiply three in the fifth power by three in the seventh, you get three in the twelfth power (5 + 7 = 12).
2. When dividing degrees having the same basis, they are left unchanged, and the indicators are subtracted. For example: when dividing three in the eighth by three to the fifth degree, you get three squared (8-5 = 3).
3. When the degree is raised to a degree, the base is left unchanged, and the indicators are multiplied. For example: when building 3 in the fifth to the seventh, you get 3 in the thirty-fifth (5x7 = 35).
4. To raise a work to a power, each of the factors is raised to the same degree. For example: when you build a 2x3 product in the fifth, you get the product two in the fifth by three in the fifth.
5. To raise a fraction to a power, the numerator and denominator are raised to the same power. For example: when 2/5 is raised to the fifth, a fraction is obtained, the numerator of which is two in the fifth, and the denominator is five in the fifth.
The noted properties of the degree are also valid for fractional indicators.
Rational exponent properties
We introduce some definitions. Any real number other than 0 raised to zero is equal to one.
Any real number other than 0 raised to a power with a negative integer is a fraction with a numerator of one and a denominator equal to the power of the same number, but with the opposite value.
We supplement the properties of the degree with several new ones that relate to rational indicators.
A degree with a rational indicator does not change when multiplying or dividing the numerator and the denominator of its indicator by an unequal zero the same number.
At the base of more than one:
- if the indicator is positive, then the degree is greater than 1;
- if negative, less than one.
With a base less than one, vice versa:
- if the indicator is positive, then the degree is less than one;
- if negative - more than 1.
When the exponent increases, then:
- the degree itself grows if the base is more than one;
- decreases if the base is less than unity.