What Are Exponents? A Quick Recap

Exponents (also called powers or indices) are a shorthand way of writing repeated multiplication. Instead of writing $2 \times 2 \times 2 \times 2 \times 2$ , we write $2^{5}$ .

Exponent

An exponent tells you how many times to multiply the base by itself. In the expression $a^{n}$ , $a$ is the base and $n$ is the exponent (or power/index).

Base

The number or variable that is being repeatedly multiplied. In $3^{4}$ , the base is $3$ .

So $2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$ .

Analogy

Think of an exponent like a photocopier set to "multiply". If you put one sheet in and set it to copy 5 times, you end up with $2^{5}$ — the original multiplied by itself, 5 times over.

The key parts of exponential notation:

Base → the number being multiplied
Exponent → how many times it is multiplied
Result → the value after evaluating

$base 3^{4 exponent} = 3 \times 3 \times 3 \times 3 = result 81$

The Seven Laws of Exponents

There are seven core rules that govern how exponents behave. These are not arbitrary — every single one can be proved from the definition of repeated multiplication. Mastering these laws allows you to simplify complex expressions quickly and confidently.

Law	Rule	Name
1	$a^{m} \times a^{n} = a^{m + n}$	Product Rule
2	$a^{m} \div a^{n} = a^{m - n}$	Quotient Rule
3	$(a^{m})^{n} = a^{m \times n}$	Power of a Power
4	$(ab)^{n} = a^{n} b^{n}$	Power of a Product
5	$(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$	Power of a Quotient
6	$a^{0} = 1$	Zero Exponent
7	$a^{- n} = \frac{1}{a ^{n}}$	Negative Exponent

Note

All of these laws apply to any non-zero base $a$ (and non-zero $b$ ). We avoid a base of zero in most cases because dividing by zero is undefined.

We will explore each of these laws in depth throughout these notes, with full explanations and worked examples.

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