The erection of a number in a natural power meansIts immediate repetition by its natural factor is a natural number of times. The number repeated as a factor is the basis of the degree, and the number indicating the number of identical factors is called the exponent. The result of the performed actions is the degree. For example, three in the sixth degree means the repetition of the number three in the form of a factor six times.
The basis of a degree can be any number other than zero.
The second and third powers of the number have special names. This, respectively, is a square and a cube.
The first power of a number is taken by the same number.
For positive numbers,which has a rational exponent. As everyone knows, any rational number is written in the form of a fraction, the numerator of which is an integer, the denominator is a natural number, that is, a positive integer, different from unity.
A power with a rational exponent representsthe root of a degree equal to the denominator of the exponent, and the radicand is the base of the power raised to a power equal to the numerator. For example: three in 4/5 is equal to the fifth root of the three in the fourth.
We note some properties that follow directly from the definition in question:
- any positive number is rational in the rational degree;
- The value of a power with a rational exponent does not depend on the form of its recording;
- if the base is negative, then the rational degree of this number is not defined.
With a positive foundation, the properties of the degree are true regardless of the exponent.
Properties of degree with natural exponent:
1. Multiplying degrees having the same bases, the base is left unchanged and the indicators are added. For example: multiplying three in the fifth degree by three in the seventh gives three to the twelfth degree (5 + 7 = 12).
2. When dividing degrees having the same bases, they are left unchanged, and the figures are subtracted. For example: if you divide three in the eighth by three in the fifth degree, you get three in a square (8-5 = 3).
3. When the degree is raised to the power, the base is left unchanged, and the indicators are multiplied. For example: when you erect 3 in the fifth to the seventh get 3 in the thirty-fifth (5x7 = 35).
4. In order to raise a product to a power, each of the factors is also built in the same way. For example: when you erect a 2x3 product in a fifth, you get a product of two in the fifth by three in the fifth.
5. To build a fraction to the power, the numerator and denominator are raised to the same degree. For example: when erecting 2/5 in a fifth, a fraction is obtained, in the numerator of which - two in the fifth, in the denominator - five in the fifth.
The noted properties of the degree are also valid for fractional exponents.
Properties of a power with rational exponent
We introduce some definitions. Any nonzero real number, raised to zero, is equal to one.
Any nonzero real number,raised to a power with a negative integer exponent is a fraction with a numerator of unity and a denominator equal to the degree of the same number but having the opposite exponent.
We supplement the properties of the degree by several new ones that relate to rational exponents.
A power with a rational exponent does not change when the numerator and denominator of its exponent are multiplied or divisible by one and the same number that is not equal to zero.
At the base more than one:
- if the indicator is positive, then the degree is greater than 1;
- at negative - less than one.
At the base less than one, on the contrary:
- if the indicator is positive, then the degree is less than one;
- at negative - more than 1.
When the exponent is increasing, then:
- the degree itself grows if the base is greater than one;
- Decreases if the base is less than one.