Surds
Surds (Extended)
- A surd is an irrational root, e.g. $\sqrt5$ — leave it exact.
- Rules: $\sqrt{a}\times\sqrt{b} = \sqrt{ab}$ and $\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt a}{\sqrt b}$.
- Simplify by taking out the largest square factor: $\sqrt{20} = \sqrt{4\times5} = 2\sqrt5$.
Practice
Simplify √20 into the form k√5. What is k?
√20 = √(4×5) = √4 × √5 = 2√5, so k = 2.
Practice
Which surd rule is correct?
√a × √b = √(ab); roots do not add like √a + √b = √(a+b).
Rationalising
- To rationalise the denominator, remove the surd from the bottom.
- $\dfrac{10}{\sqrt5} = \dfrac{10}{\sqrt5}\times\dfrac{\sqrt5}{\sqrt5} = \dfrac{10\sqrt5}{5} = 2\sqrt5$.
Practice
Rationalise 10/√5 into the form k√5. What is k?
10/√5 × √5/√5 = 10√5/5 = 2√5, so k = 2.
You've got it
Key idea
- a surd is an irrational root; keep it exact
- simplify by the largest square factor: $\sqrt{20} = 2\sqrt5$
- rationalise by multiplying top and bottom to clear the surd