These are selected revision notes and practice questions. No original exam-year or paper references have been verified; these questions are not presented as authenticated past papers. Use your current textbook and teacher guidance for complete coverage.

1. Real Number Classification Hierarchy

Real Numbers ($\mathbb{R}$)

→

Rationals ($\mathbb{Q} = p/q, q \neq 0$)

→

Irrationals ($\sqrt{2}, \pi$, Non-terminating Non-repeating)

2. Key Formulas & Theorem Summary

Master Rules Laws of Real Exponents

$a^m \cdot a^n = a^{m+n}$
$(a^m)^n = a^{m \cdot n}$
$\frac{a^m}{a^n} = a^{m-n}$
$a^0 = 1 \quad (a \neq 0)$
$a^{-n} = \frac{1}{a^n}$
$a^m \cdot b^m = (a \cdot b)^m$

2.1 Rationalization of Denominator

To eliminate surds (square roots) from the denominator, multiply both numerator and denominator by the conjugate of the denominator.

Example: Rationalize $\frac{1}{\sqrt{7} - \sqrt{6}}$

$$\frac{1}{\sqrt{7} - \sqrt{6}} \times \frac{\sqrt{7} + \sqrt{6}}{\sqrt{7} + \sqrt{6}} = \frac{\sqrt{7} + \sqrt{6}}{(\sqrt{7})^2 - (\sqrt{6})^2} = \frac{\sqrt{7} + \sqrt{6}}{7 - 6} = \sqrt{7} + \sqrt{6}$$

3. Selected Revision Practice Questions

PRACTICE Q1: Express $0.\overline{235}$ in the form $\frac{p}{q}$. (3 Marks)

Solution: Let $x = 0.235235235\dots$ (Eq 1).
Since 3 digits repeat, multiply both sides by $1000$:
$1000x = 235.235235\dots$ (Eq 2).
Subtracting Eq 1 from Eq 2:
$999x = 235 \implies x = \frac{235}{999}$.

1. Real Number Classification Hierarchy

2. Key Formulas & Theorem Summary

2.1 Rationalization of Denominator

3. Selected Revision Practice Questions

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