← Back to Grade 10 Chapters

1. Rules of Surds

A surd is an irrational number expressed with a radical symbol. Pure surds can be manipulated using these core algebraic properties:

√a × √b = √(ab)
√a / √b = √(a/b)
(√a + √b)(√a - √b) = a - b

2. Conjugate Rationalization

To eliminate radicals from a denominator, you multiply both the numerator and the denominator by the denominator's mathematical conjugate pair.

Conjugate Rule: The conjugate of (√a + √b) is simply (√a - √b). Multiplying these pairs forces an (x+y)(x-y) = x² - y² difference-of-squares distribution that removes the roots entirely.

3. Solving Radical Equations

Equations involving variable terms enclosed inside radicals require isolating the radical group before clearing it.

{/* Reason: Squaring terms before isolating radical structures creates cross-multiplication roots that make the equation harder to solve. */} Rearrange the equation terms so that the complex radical block sits completely by itself on one side of the equals sign. Square both sides of the equation to clear the radical house. Note: (√x)² = x. Isolate and solve the resulting linear or quadratic equation using standard algebra methods. Substitute your answers back into the original equation. Squaring equations can introduce fake answers called extraneous roots that do not actually work.