Extended. A surd is a root that is not a whole number, such as \(\sqrt{2}\). Instead of rounding it, simplify it by taking out square factors, collect like surds, and clear surds from the bottom of a fraction.
Use \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\) with \(a\) the largest square factor: \(\sqrt{20} = \sqrt{4}\sqrt{5} = 2\sqrt{5}\). To rationalise, multiply top and bottom by \(\sqrt{a}\), or by the conjugate: \((\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2\).
Fig. 1: a right-angled triangle with sides 4 and 2 (drawn to scale). Its hypotenuse is exactly √20 = 2√5, about 4.47
Section 01
Simplifying surds.
In the lab, choose Simplify and move the slider through numbers such as 20, 72 and 200. Watch the lab split each number into a square factor times what is left, then take the square root of the square part outside.
Rules: \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) and \(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\). Also \(\sqrt{a} \times \sqrt{a} = a\).
\(\sqrt{72} = 2\sqrt{18}\) is true but NOT fully simplified, because 18 still has the square factor 9. Use the LARGEST square factor, or keep going until no square factor is left.
Section 02
Adding, subtracting and expanding.
Like surds collect like algebra: \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\), just as \(3x + 5x = 8x\). Simplify each surd first so you can see which ones are alike. In the lab, try the Add and subtract mode.
\(\sqrt{200} - \sqrt{32}\) is NOT \(\sqrt{168}\). You can only subtract surds once they are the same surd, like \(10\sqrt{2} - 4\sqrt{2}\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Simplify \(\sqrt{50}\).
Quick check+10 XP
Work out \(\sqrt{3} \times \sqrt{12}\).
Quick check+10 XP
Rationalise \(\dfrac{6}{\sqrt{3}}\).
Quick check+10 XP
Simplify \(3\sqrt{2} + 5\sqrt{2}\).
Section 03
Rationalising the denominator.
An answer with a surd on the bottom is not in its simplest form. Multiply the top and the bottom by the same thing, chosen so that the bottom becomes a whole number. The lab's Rationalise mode shows each step.
Worked example A single surd on the bottom
Rationalise the denominator of \(\dfrac{10}{\sqrt{5}}\).
Multiply top and bottom by \(\sqrt{5}\): \(\dfrac{10\sqrt{5}}{\sqrt{5}\sqrt{5}} = \dfrac{10\sqrt{5}}{5}\).
Cancel: \(\dfrac{10}{5} = 2\).
Answer: \(2\sqrt{5}\)
Worked example Use the conjugate
Rationalise the denominator of \(\dfrac{1}{\sqrt{3} - 1}\).
Multiply top and bottom by the conjugate \(\sqrt{3} + 1\) (change the sign in the middle).
Multiplying \(\dfrac{1}{\sqrt{3} - 1}\) by \(\dfrac{\sqrt{3} - 1}{\sqrt{3} - 1}\) does not help: the bottom becomes \(4 - 2\sqrt{3}\), still a surd. Use the conjugate, with the opposite sign.
Stage 2 · Exam
Exam-style questions
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Finish the four checks above to unlock the exam questions
Exam style+20 XP
Simplify \(\sqrt{200} - \sqrt{32}\).
Exam style+20 XP
Expand and simplify \((2 + \sqrt{3})^{2}\).
Exam style+20 XP
Rationalise the denominator of \(\dfrac{1}{\sqrt{3} - 1}\).
Exam style+20 XP
Write \(\dfrac{4}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\).
Skill unlocked
Surds, sorted.
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