Surd Rules (Core)

Statement

Surds are square roots (or other roots) that cannot be simplified into whole numbers. These rules allow us to manipulate surds correctly:

\[ \sqrt{ab} = \sqrt{a}\sqrt{b}, \quad \sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}, \; b>0, \quad (\sqrt{a})^2 = a \]

Why it’s true

• Multiplication rule: squaring \(\sqrt{ab}\) gives \(ab\), and squaring \(\sqrt{a}\sqrt{b}\) also gives \(ab\), so they must be equal.
• Division rule: \(\sqrt{\tfrac{a}{b}}\) squared gives \(\tfrac{a}{b}\), while \(\tfrac{\sqrt{a}}{\sqrt{b}}\) squared also gives \(\tfrac{a}{b}\).
• Squaring a square root simply cancels the square root, returning the original number.

Recipe (how to use it)

1. Factorise the number under the root to look for perfect squares.
2. Use multiplication and division rules to break surds apart.
3. Simplify by cancelling squares when possible.

Spotting it

These rules appear in problems asking you to simplify roots, rationalise denominators, or expand expressions with surds.

Common pairings

• Expanding brackets with surds.
• Rationalising denominators.
• Indices and powers combined with surds.

Mini examples

1. Given: Simplify \(\sqrt{50}\). Answer: \(\sqrt{25 \times 2} = 5\sqrt{2}\).
2. Given: Simplify \(\sqrt{\tfrac{9}{16}}\). Answer: \(\tfrac{3}{4}\).

Pitfalls

• Forgetting that only positive \(b\) is allowed in the denominator rule.
• Leaving surds unsimplified (e.g. \(\sqrt{50}\) instead of \(5\sqrt{2}\)).
• Mixing up rules for addition (cannot simplify \(\sqrt{a}+\sqrt{b}\) unless values are equal).

Exam strategy

• Always look for square factors inside the surd.
• Apply the rules systematically — multiplication/division first, then simplify.
• Check whether the final form is in simplest surd form.

Summary

The surd rules \(\sqrt{ab}=\sqrt{a}\sqrt{b}, \sqrt{a/b}=\sqrt{a}/\sqrt{b}, (\sqrt{a})^2=a\) help to simplify radicals, handle fractions under roots, and cancel squares. They are the building blocks for more advanced algebra with surds.