What you need to know
Surds keep irrational values exact. The key skills are simplifying square roots, combining like surds and rationalising denominators without changing the value of the expression.
- Extract perfect-square factors from radicals.
- Only like surds can be added or subtracted directly.
- Use a conjugate when the denominator has two terms involving a surd.
- Keep answers exact unless a decimal approximation is requested.
Key mathematics
Product rule
\sqrt{ab}=\sqrt a\sqrt b
Use for non-negative a and b in the real-number setting.Conjugate identity
(a+b\sqrt c)(a-b\sqrt c)=a^2-b^2c
The surd terms cancel.Worked example
Simplify and rationalise $\frac{3}{2+\sqrt5}$. Multiply by the conjugate: $$\frac{3}{2+\sqrt5}\cdot\frac{2-\sqrt5}{2-\sqrt5}=\frac{3(2-\sqrt5)}{4-5}=3\sqrt5-6.$$
EXAM TECHNIQUE
Exam tips
- Simplify every surd before deciding whether terms are like terms.
- When rationalising a two-term denominator, multiply both numerator and denominator by the conjugate.
- Check the sign of the denominator after using a conjugate.
WATCH OUT
Common mistakes
- Adding $\sqrt2+\sqrt3$ as $\sqrt5$.
- Rationalising only the denominator without multiplying the numerator by the same expression.
- Turning exact surd answers into rounded decimals too early.
Frequently asked questions
Why keep surds instead of decimals?
Surds preserve exact values and are often required in algebraic and geometry answers.
What is a conjugate?
For an expression such as a plus b root c, the conjugate changes the sign between the two terms.
Can unlike surds be combined?
Not directly. Simplify them first; they combine only if they become the same surd.