The simplest mathematical expressions became known to people in ancient times. At the same time, both the operations themselves and their recordings on one or another medium were constantly being improved.
In particular, in Ancient Egypt, whose scientists made a noticeable contribution both to the development of elementary arithmetic and to the creation of the foundations of algebra and geometry, they paid attention to the fact that when a number is multiplied by the same number many times, then a huge amount of unnecessary effort is spent on it. Moreover, such an operation led to significant financial costs: according to the then prevailing settings for registration of any records, each action with a number should be described in detail. If you recall that even the simplest papyrus cost a very impressive amount of money, you should not be surprised at the efforts that the Egyptians made to find a way out of this situation.
The solution was found by the famous Diophantus of Alexandria, who came up with a special mathematical sign, which began to show how many times it is necessary to multiply one or another number by itself. Subsequently, the famous French mathematician R. Descartes improved the spelling of this expression, suggesting when designating the degree of numbers, simply ascribe it in the upper right corner above the main number.
The final chord in writing the degree of numbers was the activity of the notorious N. Schüke, who introduced into the scientific circulation first a negative, and then a zero degree.
What does the phrase “raise a degree” mean? First you need to understand that raising a power in itself is one of the most important binary mathematical operations, the essence of which is to repeatedly multiply a number by itself.
In general, this operation is indicated by the expression "XY". In this case, “X” will be called the basis of the degree, and “Y” will be called its indicator. In this case, "raise to the power" can be deciphered as "multiply" X "by itself" Y "times."
The degrees of numbers, like most other mathematical elements, have certain properties:
1. When raising to the power of zero any number other than zero (both positive and negative) will get one.
x ^^ 0 = 1
2. Degrees of numbers where indicators have a negative value should be converted to an expression with a positive indicator
xa = 1 / x ^ a
3. In order to carry out the multiplication of numbers with degrees, it should be remembered that this operation is possible only if they have the same basis. Moreover, the multiplication of numbers with degrees is carried out in accordance with the following rule: the base remains unchanged, and the value of the indicators of the other degrees is added to the index of one.
x ^ yx ^ z = x ^ y + z
4. In the case when the division of degrees occurs, it is necessary to adhere to the same rule, only instead of the amount in the indicator there will be a difference.
x ^ y / x ^ z = x ^ yz
5. Another important property of degrees is connected with those situations when it is required to raise the exponent itself to a degree. In this case, it is necessary to multiply both of these indicators.
(x ^ y) ^ z = x ^ yz
6. In some cases, there is a need to paint the degree of the product through the degree of numbers. In this case, it must be borne in mind that the degree of the product is calculated in accordance with this rule:
(xyz) ^ a = x ^ ay ^ az ^ a
7. If it becomes necessary to describe the degree of quotient, the first thing to pay attention to is that the base of the denominator cannot be equal to zero. Otherwise, you must adhere to the following formula:
(x / y) ^ a = x ^ a / y ^ a
Certain difficulties are encountered when it is necessary to raise to a power a base whose expression is less than zero. The result in this case can be both negative and positive. It will depend on the exponent, namely, on what number - odd or even - this indicator was.