# 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: $x^y$. Sometimes it is not possible. Then people write powers using the ^ sign: 2^3 means $2^3$.

The number $x$ is called base, and the number $y$ is called exponent. For example, in $2^3$, 2 is the base and 3 is the exponent.

To calculate $2^3$ a person must multiply the number 2 by itself 3 times. So $2^3=2 \cdot 2 \cdot 2$. The result is $2 \cdot 2 \cdot 2=8$. The equation could be read out loud in this way: 2 raised to the power of 3 equals 8.

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 inverse of the base. So

$x^{-1}=\frac{1}{x}$

If the exponent is an integer and is less than 0 then the person must invert 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=$\frac{3}{2}$. This is the same if you have 2 · x=3: You also get x=$\frac{3}{2}$. 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 = $\sqrt{3}$. 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).