An index (or exponent) tells you how many times a base is used as a factor. For example, \(2^4 = 2 \times 2 \times 2 \times 2\). The laws of indices let us simplify expressions efficiently and work with very large or very small quantities.
| Expression | Meaning | Value |
|---|---|---|
| \(5^3\) | \(5 \times 5 \times 5\) | \(125\) |
| \(a^2\) | \(a \times a\) | depends on \(a\) |
| \(10^0\) | zero index | \(1\) |
Index laws (for non-zero base where required)
\[ a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn} \] \[ (ab)^n = a^n b^n, \quad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \quad a^0 = 1 \]
A negative index means reciprocal: \[ a^{-n} = \frac{1}{a^n}, \qquad a \neq 0. \]
Example. \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).
Example. \(x^{-2}y^3 = \frac{y^3}{x^2}\).
Fractional indices represent roots: \[ a^{1/n} = \sqrt[n]{a}, \qquad a^{m/n} = \sqrt[n]{a^m}. \]
Example. \(27^{1/3} = 3\).
Example. \(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\).
Example 1. Simplify \(a^5 \cdot a^{-2}\).
\(a^{5+(-2)} = a^3\).
Example 2. Simplify \(\dfrac{3x^4y^{-1}}{6x^2}\).
\[ \frac{3x^4y^{-1}}{6x^2} = \frac{1}{2}x^{4-2}y^{-1} = \frac{x^2}{2y}. \]
Example 3. Simplify \((2a^3b^{-2})^2\).
\(2^2 a^{6} b^{-4} = \dfrac{4a^6}{b^4}\).
Indices are central to scientific notation: \[ N = k \times 10^n, \qquad 1 \leq k < 10. \] Positive \(n\) gives large numbers, negative \(n\) gives small decimals.
Example. \(0.00045 = 4.5 \times 10^{-4}\).