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Mathematics · Chapter 2: Real Numbers
Surds and Rationalising the Denominator
Simplifying, adding and multiplying surds, and rationalising denominators with the conjugate.
Key termRememberExampleWatch out
01Concept
A surd is a root of a rational number that cannot be simplified to a rational number. √2, √5, ∛7 are surds. √9 = 3 and ∛8 = 2 are not surds. Surds are irrational, so we keep them in surd form to stay exact instead of rounding.
In ⁿ√a, n is the index (order) of the surd and a is the radicand. In ∛5, the index is 3 and the radicand is 5. A square root has index 2, which we do not write.
Rules for surds
- √a × √b = √(ab) √3 × √5 = √15
- √a ÷ √b = √(a/b) √20 ÷ √5 = √4 = 2
- √a × √a = a √7 × √7 = 7
Simplifying a surd
Look for the largest perfect-square factor of the number under the root.
- √50 = √(25 × 2) = 5√2
- √72 = √(36 × 2) = 6√2
- √48 = √(16 × 3) = 4√3
Adding and subtracting surds
You can only add or subtract like surds (same radicand), just as you add 3x + 5x = 8x.
- 3√2 + 5√2 = 8√2
- √12 + √27 = 2√3 + 3√3 = 5√3 (simplify first!)
- √2 + √3 cannot be simplified.
Multiplying brackets with surds
- √3(2 + √3) = 2√3 + 3
- (2 + √5)(3 − √5) = 6 − 2√5 + 3√5 − 5 = 1 + √5
- (√7 + 2)(√7 − 2) = 7 − 4 = 3 (difference of two squares: the surds vanish!)
Rationalising the denominator
A fraction is in its neatest form when there is no surd in the denominator. Removing it is called rationalising the denominator.
| Denominator | Multiply top and bottom by | Example |
|---|---|---|
| √a | √a | 6/√3 = 6√3/3 = 2√3 |
| a + √b | its conjugate a − √b | 1/(3 + √2) = (3 − √2)/(9 − 2) = (3 − √2)/7 |
| a − √b | a + √b | 4/(√5 − 1) = 4(√5 + 1)/(5 − 1) = √5 + 1 |
02Key Formulas
Product of surds
Quotient of surds
Rationalising with the conjugate
Multiply top and bottom by the conjugate.
03Real World Examples
Exact answers in trigonometry
sin 45° = 1/√2 = √2/2 and sin 60° = √3/2. Engineers keep these surd forms so that errors do not build up from rounding.
Paper sizes
An A4 sheet's long side is √2 times its short side. Fold it in half and you get A5 with the same shape.
Diagonal of a plot
A square plot of side 20 m has a diagonal of 20√2 m (about 28.3 m), useful when laying a boundary wire.
04Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Simplify √98.Show answer
Answer
√98 = √(49 × 2) = 7√2.
Q2.Simplify √18 + √32 − √8.Show answer
Answer
3√2 + 4√2 − 2√2 = 5√2.
Q3.Expand and simplify (3 + √2)².Show answer
Answer
(3 + √2)² = 9 + 6√2 + 2 = 11 + 6√2.
Q4.Rationalise the denominator: 10/√5.Show answer
Answer
10/√5 × √5/√5 = 10√5/5 = 2√5.
Q5.Rationalise: 2/(√3 + 1).Show answer
Answer
- Multiply top and bottom by (√3 − 1).
- Denominator: (√3)² − 1² = 3 − 1 = 2.
- 2(√3 − 1)/2 = √3 − 1.
Q6.Is ∛27 a surd? Is √0.25?Show answer
Answer
No to both: ∛27 = 3 and √0.25 = 0.5 are rational.
Long questions
Q7.Simplify (5 + 2√3)/(2 − √3), giving your answer in the form a + b√3.Show answer
Answer
- Multiply top and bottom by the conjugate (2 + √3).
- Denominator: 2² − (√3)² = 4 − 3 = 1.
- Numerator: (5 + 2√3)(2 + √3) = 10 + 5√3 + 4√3 + 2 × 3 = 16 + 9√3.
- Answer: 16 + 9√3.