Construction of a natural power means its own direct repetition of a factor in the natural number of times.The number, repeated as a factor - a foundation degree, a number indicating the number of the same factors, called the exponent.The result of actions taken, and have a degree.For example, three in the sixth degree of repetition is a factor of three to six times.
basis of the degree can be any number other than zero.
second and third degree numbers have special names.It is, accordingly, a square and cube.
For the first power of taking the very same that number.
For positive numbers also determine the degree of having a rational figure.As we all know, any rational number written as a fraction, the numerator of which is the whole, the denominator - the natural, that is a positive integer other than one.
degree with rational exponent is a root of degree equal to the denominator of the exponent and radical expression - is the foundation degree, raised to a power equal to the numerator.For example: three in 4/5 equal to the root of the fifth degree of the three in the fourth.
note some properties, arising directly from the definition under consideration:
- any positive number in a rational degree - positive;
- importance degree with rational exponent does not depend on the form of his records;
- if the base is negative, the degree of rational numbers is not defined.
At the basis of the positive properties of the true extent independently of the indicator.
Properties degree with a natural indicator:
1. Multiplying the extent having the same base, the base is left unchanged and placed figures.For example, when multiplied by three to five degrees for three in the seventh to twelfth receive three degrees (the 5 + 7 = 12).
2. When dividing powers with the same base, they are left unchanged, and the indicators proofread.For example, when divided into three three-eighth to fifth-degree obtained three squared (8-5 = 3).
3. When the level is raised, the base is left unchanged, and the figures are multiplied.For example, in the construction of 3 in the fifth to seventh get 3 in the thirty-fifth (5x7 = 35).
4. To build the product to the extent that the same is raised each of the factors.For example, during the construction works in the 2x3 product obtained two fifth in the fifth at three in the fifth.
5. To build a power fraction in the same degree of erecting the numerator and denominator.For example, in the construction of 2/5 in the fifth receive a fraction, the numerator of which - two in the fifth, the denominator - five in the fifth.
These properties hold for the degree of fractional exponents.
Properties degree with rational exponent
introduce some definitions.Anyone other than 0 real number raised to the zero is equal to one.Any
than 0 a real number to the power of negative integer index - a fraction with the numerator and denominator of a unit equal to the degree of the same number, but having the opposite figure.
supplement the properties of several new degree, which relate to rational exponents.
degree with rational exponent does not change when multiply or divide the numerator and denominator of its indicators on unequal to zero the same number.
When the basis is greater than one:
- if the figure is positive, the level of more than 1;
- at negative - less than one.
At the basis of less than one, on the contrary:
- if the figure is positive, the level of less than one;
- at negative - more 1.
When the exponent increases, then:
- degree grows by itself, if the base is greater than one;
- decreases if the base is less than unity.