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# How to raise a number to a power?

If we go back to number rows where we considered triangular and square numbers, we can easily see that along with regular relations, including addition operations, there are regular relations based on multiplication . Let's return to the article “ Area concept “Where we got acquainted with how to determine the area of ​​a square. I hope you remember that the square of a square with a side equal to 1 (for example, one centimeter, one meter, or any other unit of measure for length) is 1x1, that is, unit of area, one square centimeter, one square meter or square of any other unit lengths The area of ​​a square with side 2 is 2 × 2 = 4. Now, if we consider squares with sides equal to 1, 2, 3, 4, 5, 6, 7, and so on, their areas will be equal to 1, 4, 9, 16, 25, 36, 49, and so on. .

Before us is a series of square numbers, which is not recorded in the form of addition 1, 1 + 3, 1 + 3 + 5, 16, 1 + 3 + 5 + 7 and so on, but as a product of 1x1, 2x2, 3x3, 4 × 4 , 5 × 5, 7x7 and so on.

Now consider a cube, that is, a three-dimensional shape that has a length, width, and height, all of which are equal to each other. An example of cubes for you can be cubes for some board game or dice. The volume of the cube is calculated by multiplying the length, width and height. This can be proved using the same technique that we used, calculating the area of ​​a square or rectangle when we multiplied the length and width.

The volume of a cube with a side equal to one is equal to one cubic unit (1x1x1 = 1), respectively. The volume of a cube with a side equal to 2 is 2x2x2 = 8, respectively, or eight cubic units. It is possible to continue such calculations, and then we get that the volume of cubes with sides 1, 2, 3, 4, 5, 6, 7, and so on is equal to 1, 8, 27, 64, 125, 216, and so on, respectively. These numbers can be represented as 1x1x1; 2x2x2; 3x3x3; 4x4x4; 5x5x5; 6x6x6i and so on.

Both squares and cubes are easy to imagine, since we often meet such figures in everyday life. But you can move away from the geometric representations and make a numerical series , where each number is the product of four, five, or six, or any other number of identical factors.

Sequential multiplication of the same number by itself is an operation that is very often used in mathematics. At one time, when we considered repeated multiple addition operations, we introduced a new concept and a new mathematical operation - multiplication. For example, we replaced 6 + 6 + 6 + 6 with 6x4. Similarly, the frequently used 6x6x6x6 multiplication operation can be briefly written down using a new symbol, a power expression: 64.

What does 64 mean? Only that we multiply the number 6 by itself four times, or 6x6x6x6. The number 105 is 10x10x10x10x10, and З2 is 3 × 3.

You can write a series of squares of numbers (12, 22, З2, 42, 52, 62, 72, and so on) and a series of cubes of numbers (13, 23, З3, 43, 53, 63, 73, and so on).

The number that is typed in small print on the upper right of the main number is called the exponent, or exponent . The number containing the exponent is called the exponential number . The number that is raised to a power, that is, multiplied by itself, is called the base of the exponential number . In expression 64, the number 6 is the base, 4 is the exponent.

Repeated multiplication of a number by itself is called raising a number to a power .

So, 64 is six to the fourth degree, similarly 105 is ten to the fifth degree. You can also say simply: six in the fourth or ten in the fifth. 32 and 33 can be called as three in the second or three in the third, but more often, following the Greek tradition, they are called three in a square or three in a cube. You can also use a table of squares and cubes of natural numbers in algebra from 1 to 100 .

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