Writing indices and index laws

In a nutshell

Indices are used as shorthand for repeated multiplication. Another term for indices is "powers".

Writing indices

Indices are written as a small number to the top right of another number. The index number (also called the power) tells you the number of times you multiply the bigger number by itself.

Examples

$5^4$ is shorthand for $5\times5\times5\times5=625$ . It is read as "five to the power of four".
$15^2$ is shorthand for $15\times15=225$ . Anything to the power of two is called "squared". In this case, this is read as "fifteen squared".
$1^3$ is shorthand for $1\times1\times1=1$ . Anything to the power of three is called "cubed". In this case, this is read as "one cubed".

Index laws

Index laws (or the laws of indices) are a set of rules that help with writing powers or multiplying/dividing with them. Here are the rules:

The six index laws

When multiplying two powers of the same number, add the powers.
When dividing two powers of the same number, subtract the powers.
When raising a power to another power, multiply the powers.
A negative power is the same as turning the number "upside down" like a fraction.
Anything to the power of $1$ is itself.
Anything to the power of $0$ is $1$ .

Examples

$6^3\times6^5=6^{3+5}=6^8$
$13^7\div13^{-2}=13^{7-(-2)}=13^9$
$(2^8)^3=2^{8\times3}=2^{24}$
$3^{-5}=\frac{1}{3^5}$
$33^1=33$
$33^0=1$