- 1. What Makes a Surd — Rational, Irrational, and the Rules
Distinguish rational from irrational numbers, test whether a root is a surd by checking the radicand for a perfect power, apply the product rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and quotient rule $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$, and explain with a counter-example why $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$.
- 2. Simplifying Surds to Basic Form
Define basic (simplest) form, find the largest perfect-square factor of a radicand, use the product rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ to extract it and write the surd as a coefficient times a smaller surd, and recognise and complete an unfinished simplification.
- 3. Entire Surds and Comparing Surds
Convert a mixed surd $a\sqrt{b}$ into a single (entire) surd $\sqrt{a^2 b}$ by squaring the coefficient, convert back to basic form, and compare or order surds without a calculator by writing each as an entire surd and comparing radicands.
- 4. Like Surds — Addition and Subtraction
Identify like (similar) surds, add and subtract them by combining coefficients while keeping the root unchanged, simplify every surd to basic form first because like surds arrive disguised, and handle expressions with two or more surd families, keeping each family separate.
- 5. Multiplying and Dividing Surds
Multiply surds with $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ (coefficients times coefficients, radicands times radicands), divide with $\sqrt{a} \div \sqrt{b} = \sqrt{a/b}$, apply the key fact $\sqrt{a} \times \sqrt{a} = a$, and always simplify the result to basic form — including products and quotients that collapse into whole numbers.
- 6. Expanding Brackets and Conjugate Surds
Expand a single bracket of surds by distribution, expand two surd binomials with FOIL, square a surd binomial without dropping the cross term, and recognise the conjugate pattern $(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = x - y$ — the difference-of-squares cancellation that always leaves a rational number.
- 7. Rationalising a Single-Surd Denominator
Rationalise a fraction with a single surd in the denominator by multiplying numerator and denominator by that surd, using $\sqrt{a} \times \sqrt{a} = a$ to clear the root and $\dfrac{\sqrt{a}}{\sqrt{a}} = 1$ to keep the value unchanged. Simplify the denominator to basic form first where needed, and reduce the final fraction.
- 8. Rationalising with Conjugates
Clear a two-term surd denominator by multiplying numerator and denominator by the denominator's conjugate, so the bottom becomes a difference of squares $(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y}) = x - y$. FOIL the numerator, simplify, and express the answer in a required named form such as $x + y\sqrt{2}$ or $p\sqrt{5} + q\sqrt{3}$, watching signs.
- 9. Equality of Surds
Use the principle that if $p + m\sqrt{k} = q + n\sqrt{k}$ (with $\sqrt{k}$ irrational) then $p = q$ and $m = n$, to solve for unknown coefficients and to find the square root of an expression of the form $a + b\sqrt{c}$ by setting it equal to $\sqrt{m} + \sqrt{n}$, squaring, and equating the rational and surd parts.
- 10. Surds in Trigonometry — The Capstone
Build the exact surd values of sin, cos, and tan for $30°$, $45°$, and $60°$ from the 45-45-90 and 30-60-90 triangles, state the exact-value table, simplify and rationalise trigonometric expressions involving surds, and solve right-triangle problems leaving the answer in exact surd form.
