Indices (also called powers or exponents) are a shorthand way of writing repeated multiplication. For example, $a^3$ means $a \times a \times a$. The laws of indices are rules that let you combine and simplify expressions involving powers without expanding them.

The Key Laws

For a base $a$ (where $a \neq 0$) and integers $m$ and $n$:

• Multiplication law: $a^m \times a^n = a^{m+n}$
• Division law: $\dfrac{a^m}{a^n} = a^{m-n}$
• Power of a power: $(a^m)^n = a^{mn}$

These laws only apply directly when the bases are the same. If the bases differ, you may need to rewrite one of them first.

We also have:

• Power of a product: $(ab)^n = a^n b^n$
• Power of a quotient: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

Why the Laws Work

The multiplication law follows from counting factors:

The division law works similarly: the factors in the denominator cancel with factors in the numerator, leaving $m - n$ factors of $a$.