What are Indices (Exponents)?
An index (plural: indices) or exponent tells us how many times a number (the base) is multiplied by itself. For example, in \(a^m\), \(a\) is the base and \(m\) is the index/exponent.
What are the Laws of Indices?
The laws of indices are rules that help us simplify expressions involving powers. They work for any real base (except 0⁰) and integer exponents.
The Five Basic Laws of Indices:
| Law | Rule | Example |
|---|---|---|
| **Division Law** | \(a^m \div a^n = a^{m-n}\) (where \(m > n\)) | \(5^6 \div 5^2 = 5^4\) |
| **Power of a Power Law** | \((a^m)^n = a^{m \times n}\) | \((3^2)^4 = 3^8\) |
| **Power of a Product Law** | \((ab)^m = a^m \times b^m\) | \((2 \times 3)^4 = 2^4 \times 3^4\) |
| **Power of a Quotient Law** | \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\) (where \(b \neq 0\)) | \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\) |
Why Do These Laws Work?
They are derived from the definition of exponents as repeated multiplication:
- \(a^m \times a^n = (a \times a \times ... \times a) \times (a \times a \times ... \times a) = a^{m+n}\)