# Exponentiation

Exponentiation (power) is an arithmetic operation on numbers. It is repeated multiplication, just as multiplication is repeated addition. People write exponentiation with upper index. This looks like this: $x^{y}$ . Other methods of mathematical notation have been used in the past. When writing with equipment that cannot use the upper index, people write powers using the ^ or ** signs, so 2^4 or 2**4 means $2^{4}$ .

The number $x$ is called base, and the number $y$ is called exponent. For example, in $2^{4}$ , 2 is the base and 4 is the exponent.

To calculate $2^{4}$ a person must multiply 4 copies of 2. So $2^{4}=2\cdot 2\cdot 2\cdot 2$ . The result is $2\cdot 2\cdot 2\cdot 2=16$ . The equation could be read out loud in this way: 2 raised to the power of 4 equals 16.

Examples:

• $5^{3}=5\cdot {}5\cdot {}5=125$ • $x^{2}=x\cdot {}x$ • $1^{x}=1$ 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 $a^{2}$ . So

$x^{2}$ is the square of $x$ If the exponent is equal to 3, then the power is called cube because the volume of a cube is calculated using $a^{3}$ . So

$x^{3}$ is the cube of $x$ If the exponent is equal to -1 then the person must calculate the reciprocal of the base. So

$x^{-1}={\frac {1}{x}}$ If the exponent is an integer and is less than 0 then the person must find the reciprocal the number and calculate the power. For example:

$2^{-3}=\left({\frac {1}{2}}\right)^{3}={\frac {1}{8}}$ If the exponent is equal to ${\frac {1}{2}}$ then the result of exponentiation is the square root of the base. So $x^{\frac {1}{2}}={\sqrt {x}}.$ Example:

$4^{\frac {1}{2}}={\sqrt {4}}=2$ Similarly, if the exponent is ${\frac {1}{n}}$ the result is the nth root, so:

$a^{\frac {1}{n}}={\sqrt[{n}]{a}}$ If the exponent is a rational number ${\frac {p}{q}}$ , then the result is the qth root of the base raised to the power of p, so:

$a^{\frac {p}{q}}={\sqrt[{q}]{a^{p}}}$ 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 (xi), whose limit is x:

$x=\lim _{n\to \infty }x_{n}$ like this:

$a^{x}=\lim _{n\to \infty }a^{x_{n}}$ There are some rules which help to calculate powers:

• $\left(a\cdot b\right)^{n}=a^{n}\cdot {}b^{n}$ • $\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}},\quad b\neq 0$ • $a^{r}\cdot {}a^{s}=a^{r+s}$ • ${\frac {a^{r}}{a^{s}}}=a^{r-s},\quad a\neq 0$ • $a^{-n}={\frac {1}{a^{n}}},\quad a\neq 0$ • $\left(a^{r}\right)^{s}=a^{r\cdot s}$ • $a^{0}=1$ It is possible to calculate exponentiation of matrices. The matrix must be square. For example: $I^{2}=I\cdot I=I$ .

## 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 x = ${\textstyle {\sqrt[{2}]{3}}}$ . However, if you have 2x=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 x=log2(3).