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- About
# Exponent calculator

In mathematics, the term "exponentiation" is widely used, that is, multiplying a number by itself. The second power is a single multiplication, the third power is a double multiplication. If the number is denoted by the letter a, the degree by the letter n, then the expression will look like a^n. In this case, a will be called the base of the degree, and n - the exponent.

## A bit of history

The definitions “number squared” and “number cubed”, which are familiar to us, arose in ancient Greece. Then they were called a little differently: “square on segment a” and “cube on a”. Greek mathematicians did not use higher degrees (above the third), and the need for them arose only during the Late Middle Ages and the New Age: as a component of scientific and technological progress.

Until the 17th century in Europe, the degree of numbers was written with letter symbols. So, the symbol q meant a square, c - a cube, qq - a bi-square, and so on. Another common form of writing was the lowercase listing of factors, for example - xx or xxx, which equaled x squared and cubed, respectively.

The French scientist Pierre Erigon and the Scottish mathematician James Hume proposed their own system - to write the degree lowercase, to the right of the base. That is, write modern a² and a³ as a2 and a3.

Experiments with displaying degrees continued until 1637, when the French scientist René Descartes introduced the now convenient form: a^n. At the same time, the indicator n until the 70s of the 17th century could be expressed by any natural numbers, except for two (square). And the square of the number continued to be written as a2.

In 1676, Isaac Newton and John Valles extended the system proposed by Descartes to include fractional and negative exponents. In 1679, numbers began to be raised to a variable degree (at the suggestion of Gottfried Leibniz), and in 1743 - to an imaginary one (at the initiative of Leonhard Euler)).

## Modern definition

Today, the definition of degrees is clearly articulated and expressed in formulas that have stood the test of time and have not changed over the past 200-250 years. According to official mathematical reference books, the power n of a number a is the product of factors of magnitude a n times in a row. Or, if we represent this description as a formula, then:

a^n = (a ⋅ a ⋅ a ⋅ ... a),

where the number of factors a after the equal sign is equal to n.

For all degrees, except for the second and third, only "nominal" names are provided: fourth, fifth, twenty-sixth, hundredth, and so on. And the second and third are also called "square" and "cube" respectively.

There is also the concept of "extensions" for powers, which include integer, rational, real, and complex powers. They are included in the sections of higher mathematics, are quite difficult to understand and are used only in complex technical calculations.

## Use in computer science

With the development of digital technologies, it became necessary to use degrees in programming languages for which there is no “two-story” display of characters in the form of a². It was replaced by "one-story" expressions: "**" and "↑". The former was used in Fortran and the latter in Algol.

The final and most common form (for the BASIC programming language and later) was the circumflex symbol "^", which today is used to display degrees not only in programming, but also when writing mathematical formulas on paper (and in text editors) . Using this symbol, the usual expression a² looks like a^2. It is also used in many models of calculators, including engineering ones.

Summing up, we can say that exponentiation is a convenient and indispensable mathematical operation, on which many formulas and physical laws are based. Its inverse operation is extracting the root.

Squaring small numbers can be done in your head, but for more complex calculations, you need either an engineering calculator or an online application with advanced features.

- Help
## Laws of exponents (exponent rules)

Algebra, geometry, trigonometry and physics are difficult to imagine without such a mathematical operation as exponentiation. It is used in many rules and formulas, for example - in the Pythagorean theorem (a² + b² = c²) or in the law of equivalence of mass and energy (E = mc²). Complex equations with several unknowns raised to the same or different powers need to be simplified first. It is based on the properties of degrees, which are worth considering in more detail.

### Artwork

Powers, like their bases, can be reduced to general values, and one of the simplest principles is the product. So, two identical numbers raised to different powers turn into one number with the summed power:

a^n ⋅ a^m = a^(n + m).

N and m can be any natural numbers. For example, if we multiply 82 by 84, we get 86. And if we multiply 1610 by 163, we get 1613. This rule also applies to unknowns (x, y, z), the powers of which can be added with an indefinite common base.

### Private

This property works “in the opposite direction” compared to the previous one, that is, when dividing the same numbers with different degrees, the latter are subtracted, and the base remains the same:

a^n / a^m = a^(n − m).

As with the product, n and m must be natural numbers, but there are two more important rules for the quotient. The first is that n must be greater than m (so that after subtraction it does not turn out to be a negative number). And secondly, a must not be equal to zero (division by 0 is prohibited).

### Exponentiation

Powers, like their bases, can also be raised to a power, and there is an unshakable rule for this:

(a^n)^m = a^(n ⋅ m).

That is, in this case, the powers are multiplied - provided that n and m are natural numbers. For example, the expression (6^5)^3 can be simplified to 6^15. The rule also works the other way around. So, 9^10 can be turned into (9^5)^2. This becomes especially useful when simplifying complex equations and trigonometric functions.

### Product to the power

The brackets can contain not one, but several numbers that are raised to the same power as follows:

(a ⋅ b)^n = a^n ⋅ b^n.

For example, the value (12 ⋅ 7)^3 can be expressed as 12^3 ⋅ 7^3. Like the previous one, this property is also reversible, and 12^3 ⋅ 7^3 can be turned back into (12 ⋅ 7)^3. The main condition remains the same - n must be a natural number.

### Partial

A mirror version of the previous property, in which the product is replaced by a quotient:

(a / b)^n = a^n / b^n.

Instead of a, you can substitute any number, and instead of b - everything except zero (due to the impossibility of dividing by 0). In turn, n must be a natural number. For example, replacing a with 8, b with 3, and n with 10, we get: (8 / 3)^10 = 8^10 / 3^10.

### Addition and subtraction

If the numbers have different bases and degrees, it will not work to bring them to a common form (as described above). In this case, you will have to work separately with degrees and separately with bases. For example, to add the numbers 22 and 34, you must first turn them into 8 and 81, and only then add them: 8 + 81 = 89.

The main thing to remember is that exponentiation comes first in precedence, multiplication and division come second, and addition and subtraction come third. If the expression contains brackets, then the calculation is carried out inside them first, and then outside them. For example, in the expression:

(x^2 ⋅ x^4) ⋅ z + h,

First of all, we add the powers of x, turning the product into a single number x^6, after which we multiply it by z, and at the end we add h.

The listed properties of degrees are not theorems and do not need proof. They are used by default in all equations and formulas, and are also included in the algorithms of all computing applications. Including - in the algorithms of special online calculators, with which you can quickly raise any number to any power by opening a tab in your browser.

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