What are Negative Indices?
A negative index indicates the reciprocal of the base raised to the positive index:
\[
a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
\]
Why Do Negative Indices Work?
Using the division law: \(a^0 \div a^n = a^{0-n} = a^{-n}\). But \(a^0 = 1\), so \(1 \div a^n = \frac{1}{a^n}\).
What are Fractional Indices?
A fractional index represents a root:
\[
a^{\frac{1}{n}} = \sqrt[n]{a} \quad \text{(nth root of a)}
\]
\[
a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m
\]
Summary of Index Types:
| Index Type | Meaning | Example | Value |
|---|---|---|---|
| Zero | Always 1 (a≠0) | \(5^0\) | \(1\) |
| Negative integer | Reciprocal | \(2^{-3}\) | \(\frac{1}{8} = 0.125\) |
| Fraction (1/n) | nth root | \(8^{\frac{1}{3}}\) | \(\sqrt[3]{8} = 2\) |
| Fraction (m/n) | Power and root | \(8^{\frac{2}{3}}\) | \((\sqrt[3]{8})^2 = 2^2 = 4\) |
Important Rules for Fractional Indices:
- \(a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m = \sqrt[n]{a^m}\)
- The order of power and root doesn't matter: \(\sqrt[n]{a^m} = (\sqrt[n]{a})^m\)