# Exponentiation

In mathematics, **exponentiation** (**power**) is an arithmetic operation on numbers. It can be thought of as repeated multiplication, just as multiplication can be thought of as repeated addition.

In general, given two numbers \({\displaystyle x}\) and \({\displaystyle y}\), the exponentiation of \({\displaystyle x}\) and \({\displaystyle y}\) can be written as \({\displaystyle x^{y}}\), and read as "\({\displaystyle x}\) raised to the power of \({\displaystyle y}\)", or "\({\displaystyle x}\) to the \({\displaystyle y}\)th power".^{[1]}^{[2]} Other methods of mathematical notation have been used in the past. When the upper index cannot be written, people can write powers using the `^` or ** signs, so that `2^4 or` 2**4 means \({\displaystyle 2^{4}}\).

Here, the number \({\displaystyle x}\) is called **base**, and the number \({\displaystyle y}\) is called **exponent**. For example, in \({\displaystyle 2^{4}}\), 2 is the base and 4 is the exponent.

To calculate \({\displaystyle 2^{4}}\), one simply multiply 4 copies of 2. So \({\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2}\), and the result is \({\displaystyle 2\cdot 2\cdot 2\cdot 2=16}\). The equation could be read out loud as "2 raised to the power of 4 equals 16."

More examples of exponentiation are:

- \({\displaystyle 5^{3}=5\cdot {}5\cdot {}5=125}\)
- \({\displaystyle x^{2}=x\cdot {}x}\)
- \({\displaystyle 1^{x}=1}\) for every number
*x*

If the exponent is equal to 2, then the power is called **square**, because the area of a square is calculated using \({\displaystyle a^{2}}\). So

- \({\displaystyle x^{2}}\) is the square of \({\displaystyle x}\)

Similarly, if the exponent is equal to 3, then the power is called **cube**, because the volume of a cube is calculated using \({\displaystyle a^{3}}\). So

- \({\displaystyle x^{3}}\) is the cube of \({\displaystyle x}\)

If the exponent is equal to -1, then the power is simply the reciprocal of the base. So

- \({\displaystyle x^{-1}={\frac {1}{x}}}\)

If the exponent is an integer less than 0, then the power is the reciprocal raised to the opposite exponent. For example:

- \({\displaystyle 2^{-3}=\left({\frac {1}{2}}\right)^{3}={\frac {1}{8}}}\)

If the exponent is equal to \({\displaystyle {\tfrac {1}{2}}}\), then the result of exponentiation is the square root of the base, with \({\displaystyle x^{\frac {1}{2}}={\sqrt {x}}.}\) For example:

- \({\displaystyle 4^{\frac {1}{2}}={\sqrt {4}}=2}\)

Similarly, if the exponent is \({\displaystyle {\tfrac {1}{n}}}\), then the result is the nth root, where:

- \({\displaystyle a^{\frac {1}{n}}={\sqrt[{n}]{a}}}\)

If the exponent is a rational number \({\displaystyle {\tfrac {p}{q}}}\), then the result is the *q*th root of the base raised to the power of *p*:

- \({\displaystyle a^{\frac {p}{q}}={\sqrt[{q}]{a^{p}}}}\)

In some cases, the exponent may not even be rational. To raise a base *a* to an irrational *x*th power, we use an infinite sequence of rational numbers (*x _{n}*), whose limit is x:

- \({\displaystyle x=\lim _{n\to \infty }x_{n}}\)

like this:

- \({\displaystyle a^{x}=\lim _{n\to \infty }a^{x_{n}}}\)

There are some rules which make the calculation of exponents easier:^{[3]}

- \({\displaystyle \left(a\cdot b\right)^{n}=a^{n}\cdot {}b^{n}}\)
- \({\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}},\quad b\neq 0}\)
- \({\displaystyle a^{r}\cdot {}a^{s}=a^{r+s}}\)
- \({\displaystyle {\frac {a^{r}}{a^{s}}}=a^{r-s},\quad a\neq 0}\)
- \({\displaystyle a^{-n}={\frac {1}{a^{n}}},\quad a\neq 0}\)
- \({\displaystyle \left(a^{r}\right)^{s}=a^{r\cdot s}}\)
- \({\displaystyle a^{0}=1}\)

It is possible to calculate exponentiation of matrices. In this case, the matrix must be square. For example, \({\displaystyle I^{2}=I\cdot I=I}\).

## Contents

## Commutativity

Both addition and multiplication are commutative. For example, 2+3 is the same as 3+2, and 2 · 3 is the same as 3 · 2. Although exponentiation is repeated multiplication, it is not commutative. For example, 2³=8, but 3²=9.

## Inverse Operations

Addition has one inverse operation: subtraction. Also, multiplication has one inverse operation: division.

But exponentiation has two inverse operations: The root and the logarithm. This is the case because the exponentiation is not commutative. You can see this in this example:

- If you have x+2=3, then you can use subtraction to find out that x=3−2. This is the same if you have 2+x=3: You also get x=3−2. This is because x+2 is the same as 2+x.
- If you have x · 2=3, then you can use division to find out that x=\({\textstyle {\frac {3}{2}}}\). This is the same if you have 2 · x=3: You also get x=\({\textstyle {\frac {3}{2}}}\). This is because x · 2 is the same as 2 · x
- If you have x²=3, then you use the (square) root to find out x: you get the result that x = \({\textstyle {\sqrt[{2}]{3}}}\). However, if you have 2
^{x}=3, then you can not use the root to find out x. Rather, you have to use the (binary) logarithm to find out x: you get the result that x=log_{2}(3).

## Related pages

## References

- ↑ "Compendium of Mathematical Symbols" .
*Math Vault*. 2020-03-01. Retrieved 2020-08-28. - ↑ Weisstein, Eric W. "Power" .
*mathworld.wolfram.com*. Retrieved 2020-08-28. - ↑ Nykamp, Duane. "Basic rules for exponentiation" .
*Math Insight*. Retrieved August 27, 2020.

**Categories:** Mathematics | Hyperoperations

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