Any arithmetic action has its opposite.
Addition is the opposite of subtraction, multiplication is division.
There are their "counterparts-antipodes" and the exponentiation.
Exponentiation assumes that this number must be multiplied by itself a certain number of times. For example, raising the number 2 to the fifth power will look like this:
2 * 2 * 2 * 2 * 2 = 64.
The number to be multiplied by itself,Is called the base of a power, and the number of multiplications is its exponent. Two opposite actions correspond to the exponentiation: finding the exponent and finding the base.
Extraction of the root
Finding the basis of a degree is called root extraction. This means that it is necessary to find a number that needs to be raised to the power of n to get this.
For example, it is necessary to extract the fourth root from the number 16, i.e. Determine what number you need to multiply by itself 4 times to eventually get 16. This number is 2.
Such an arithmetic action is recorded using a special sign - the radical:?, Above which the exponent is indicated to the left.
If the exponent is an even number,Then the root may be two numbers with the same modulus, but with different signs - positive and negative. So, in the above example, it can be the numbers 2 and -2.
The expression must be unambiguous, i.e. Have one result. For this purpose, the notion of an arithmetic root was introduced, which can represent only a positive number. To be less than zero the arithmetic root can not.
Thus, in the example considered above, the arithmetic root is only the number 2, and the second variant of the answer -2 is eliminated by definition.
For some degrees that are used more often than others, in mathematics there are special names that are initially associated with geometry. It's about erecting in the second and third degree.
In the second degree, the length of the sideSquare, when you need to calculate its area. If you want to find the volume of the cube, the length of its edges is raised to the third power. Therefore the second degree is called the square of the number, and the third is called the cube.
Correspondingly, the root of the second degree is calledSquare, and the root of the third degree - cubic. The square root is the only root of which the exponent is not put above the radical:
? 64 = 8
Thus, the arithmetic square root of a given number is a positive number that must be raised to the second power in order to obtain a given number.