Exponentiation

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

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

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

Examples:

• ${\displaystyle 5^{3}=5\cdot {}5\cdot {}5=125}$
• ${\displaystyle x^{2}=x\cdot {}x}$
• ${\displaystyle 1^{x}=1}$ 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 ${\displaystyle x}$

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 person must calculate the inverse of the base. So

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

If the exponent is an integer and is less than 0 then the person must invert the number and calculate the power. For example:

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

If the exponent is equal to ${\displaystyle {\frac {1}{2}}}$ then the result of exponentiation is the square root of the base. So ${\displaystyle x^{\frac {1}{2}}={\sqrt {x}}.}$ Example:

${\displaystyle 4^{\frac {1}{2}}={\sqrt {4}}=2}$

Similarly, if the exponent is ${\displaystyle {\frac {1}{n}}}$ the result is the nth root, so:

${\displaystyle a^{\frac {1}{n}}={\sqrt[{n}]{a}}}$

If the exponent is a rational number ${\displaystyle {\frac {p}{q}}}$, then the result is the qth root of the base raised to the power of p, so:

${\displaystyle a^{\frac {p}{q}}={\sqrt[{q}]{a^{p}}}}$

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:

${\displaystyle x=\lim _{n\to \infty }x_{n}}$

like this:

${\displaystyle a^{x}=\lim _{n\to \infty }a^{x_{n}}}$

There are some rules which help to calculate powers:

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

It is possible to calculate exponentiation of matrices. The matrix must be square. For example: ${\displaystyle I^{2}=I\cdot I=I}$.

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).