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 qth 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 xth 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, ${\displaystyle I^{2}=I\cdot I=I}$.

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