Indices and Surds Cheat Sheet

This cheat sheet covers the fundamental rules and applications of indices, surds, and scientific notation, essential for simplifying expressions and working with very large or small numbers.

Core Principles

Index laws provide shortcuts for multiplying, dividing, and raising powers to simplify expressions.
Negative indices indicate reciprocals, relating to division.
Fractional indices represent roots, allowing for the simplification of radical expressions.
Surds are irrational numbers involving roots that cannot be simplified to a rational number.
Scientific notation is a standardized way to express very large or very small numbers using powers of 10.
Significant figures are crucial for accurately representing the precision of measurements in calculations.

Formulas

$a^m \times a^n = a^{m+n}$
$a^m \div a^n = a^{m-n}$
$({a^m})^n = a^{mn}$
$a^0 = 1 \text{ (where } a \neq 0)$
$(ab)^m = a^m b^m$
$(a/b)^m = a^m / b^m$
$a^{-m} = 1/a^m$
$a^{1/n} = \sqrt[n]{a}$
$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
$\sqrt{a} \div \sqrt{b} = \sqrt{a/b}$
$a \times 10^m \text{ (where } 1 \le |a| < 10 \text{ and } m \text{ is an integer)}$