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Surds · Lesson 2 of 2

Rationalising the denominator

Removing surds from the bottom of a fraction: multiply by the surd for a single root, and by the conjugate for a sum or difference.

14 minYou should already know: Indices & standard form
  1. 1
  2. 2

An answer in surd form shouldn’t have a surd in the denominator (the bottom of the fraction). Removing it is called rationalising. You multiply the top and bottom by the same thing, so the value doesn’t change, only the way it is written. You need simplifying surds first.

One surd on the bottom

Multiply top and bottom by that surd, because a×a=a\sqrt a \times \sqrt a = a:

6√3×√3√3=6√33=2√3this is 1
Multiply by 1√3 ÷ √3 is 1, so the value doesn't change

More: one surd on the bottom

A sum or difference on the bottom: the conjugate

For 1a+b\frac{1}{a + \sqrt b}, multiplying by b\sqrt b doesn’t clear the surd. Instead, multiply by the conjugate: the same two terms with the sign between them changed. The difference of two squares then removes the surd:

(a+b)(a−b)=a2−b(a + \sqrt b)(a - \sqrt b) = a^2 - b
a− √ba+ √ba²− a√b+ a√b− bmiddle terms cancel: a² − b
The difference of two squares(a + √b)(a − √b) = a² − b: the surd disappears

Step through all three kinds and watch the decimal value: it never changes, because top and bottom are multiplied by the same thing.

Rationalising the denominatorPick a fraction, then press Next
  1. 6√3
√3multiply top and bottom by3.4641value, as a decimal
One surd on the bottom: multiply by that surd, because √3 × √3 = 3. Press Next.

Worked example · WAEC 2019

WAEC 2019 · Paper 2 · Q3 (a)

Without using mathematical tables or calculators, simplify 332−423−243\sqrt{\frac32} - 4\sqrt{\frac23} - \sqrt{24}.

  1. Rationalise each surd

    • 332=332×22=362{3\sqrt{\frac32} = \frac{3\sqrt3}{\sqrt2} \times \frac{\sqrt2}{\sqrt2} = \frac{3\sqrt6}{2}}.
    • 423=423×33=463{4\sqrt{\frac23} = \frac{4\sqrt2}{\sqrt3} \times \frac{\sqrt3}{\sqrt3} = \frac{4\sqrt6}{3}}.

    Think first. 32=32\sqrt{\frac32} = \frac{\sqrt3}{\sqrt2}. What do you multiply by?

  2. The third term

    24=4×6=26\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt6.

  3. Collect over a common denominator

    Over 6, the three terms are 966\frac{9\sqrt6}{6}, 866\frac{8\sqrt6}{6} and 1266\frac{12\sqrt6}{6}:

    966−866−1266=(9−8−12)66=−1166\begin{aligned} &\frac{9\sqrt6}{6} - \frac{8\sqrt6}{6} - \frac{12\sqrt6}{6} \\ &= \frac{(9 - 8 - 12)\sqrt6}{6} \\ &= -\frac{11\sqrt6}{6} \end{aligned}

    Think first. What common denominator do 2 and 3 need?

More: rationalising with the conjugate

Matching a form x + y√n

Some questions give the form of the answer and ask for xx and yy. Rationalise, then match the whole-number parts, and the surd parts, on the two sides.

More: matching a form

Your turn

JAMB 1997 · UME · Q7

Simplify 23+3535−23\dfrac{2\sqrt3 + 3\sqrt5}{3\sqrt5 - 2\sqrt3}.

Worked solution (try it first)
  1. Multiply the top and bottom by the conjugate of the bottom, 35+233\sqrt5 + 2\sqrt3.
  2. Bottom: (35)2−(23)2=45−12(3\sqrt5)^2 - (2\sqrt3)^2 = 45 - 12, which is 33.
  3. Top: (23+35)2=12+1215+45(2\sqrt3 + 3\sqrt5)^2 = 12 + 12\sqrt{15} + 45, which is 57+121557 + 12\sqrt{15}.
  4. Divide top and bottom by 3: 19+41511\dfrac{19 + 4\sqrt{15}}{11}, option A.

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