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:

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

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 (x_i), 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:

$\left(a\cdot b\right)^n = a^n\cdot{}b^n$
$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n},\quad b\neq 0$
$a^r \cdot{} a^s = a^{r+s}$
$\frac{a^r}{a^s} = a^{r-s},\quad a\neq 0$
$a^{-n} = \frac{1}{a^n},\quad a\neq 0$
$\left(a^r\right)^s = a^{r\cdot s}$
$a^0 = 1,\quad a\neq 0$ : Where the base is greater than 1 and the exponent is 0, the answer is 1. If the base and exponent are both 0, the answer is undefined.