Understanding Surds
A surd is an expression involving a root, typically a square root, that cannot be simplified to a rational number. For example, \( \sqrt{2} \) is a surd because it is an irrational number and cannot be expressed exactly as a fraction.
Surds retain the root symbol because their exact value is irrational, unlike \( \sqrt{4} = 2 \), which is rational and can be simplified fully.
Recognising surds is important because they often appear in exact forms in mathematics, and simplifying them helps in further calculations.
- Surds are irrational roots that cannot be simplified to remove the root sign.
- Examples: \( \sqrt{2}, \sqrt{3}, \sqrt{5} \) are surds; \( \sqrt{9} = 3 \) is not a surd.
Simplifying Surds
To simplify a surd, factor the number under the root into a product of a perfect square and another number. Then use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
Extract the square root of the perfect square as a whole number, leaving the surd with the remaining factor.
Example 1: Simplify \( \sqrt{20} \). Since 20 = 4 × 5 and 4 is a perfect square, \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \). This is the simplest form because 5 has no perfect square factors other than 1.
- Find the largest perfect square factor of the number under the root.
- Rewrite the surd as the product of the square root of the perfect square and the remaining surd.
- Simplify the square root of the perfect square to a whole number.
Simplifying Surds with Expressions Under the Root
Sometimes the expression under the root is itself a calculation. Simplify inside the root first, then simplify the surd.
Example 2: Simplify \( \sqrt{200 - 32} \). Calculate inside the root: 200 - 32 = 168.
Next, factor 168 to find the largest perfect square factor. 168 = 4 × 42, and 4 is a perfect square, so \( \sqrt{168} = \sqrt{4 \times 42} = 2\sqrt{42} \). This is the simplest form since 42 has no perfect square factors other than 1.
- Calculate the value inside the root first if it is an expression.
- Then simplify the surd by factoring out perfect squares as before.
Rationalising Denominators with Single Surds
Rationalising the denominator means rewriting a fraction so that the denominator contains no surds.
If the denominator is a single surd, multiply numerator and denominator by the same surd to remove the root from the denominator.
Example 3: Rationalise \( \frac{1}{\sqrt{10}} \). Multiply numerator and denominator by \( \sqrt{10} \): \( \frac{1}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{10}}{10} \). Now the denominator is rational.
- Multiply numerator and denominator by the surd in the denominator.
- Use the property \( \sqrt{a} \times \sqrt{a} = a \) to simplify the denominator.
- Simplify the fraction if possible.
Rationalising Denominators with Binomial Surds
When the denominator is a binomial involving surds, such as \( 1 + \sqrt{3} \), multiply numerator and denominator by the conjugate, which is the same binomial with the opposite sign between terms.
The conjugate of \( 1 + \sqrt{3} \) is \( 1 - \sqrt{3} \). Multiplying by the conjugate uses the difference of squares formula: \( (a + b)(a - b) = a^2 - b^2 \).
Example 4: Rationalise \( \frac{1}{1 + \sqrt{3}} \). Multiply numerator and denominator by \( 1 - \sqrt{3} \): \[ \frac{1}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{1 - \sqrt{3}}{1 - 3} = \frac{1 - \sqrt{3}}{-2} = -\frac{1 - \sqrt{3}}{2} \]. The denominator is now rational.
- Identify the conjugate of the denominator.
- Multiply numerator and denominator by the conjugate.
- Simplify the denominator using the difference of squares formula.
- Simplify the resulting expression carefully.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Simplifying Surds | Factor number under root into perfect square × other number | \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \) | Always simplify surds fully before further calculations |
| Rationalising Single Surd Denominator | Multiply numerator and denominator by the surd | \( \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10} \) | Removes surds from denominator for exact answers |
| Rationalising Binomial Surd Denominator | Multiply numerator and denominator by conjugate | \( \frac{1}{1 + \sqrt{3}} = -\frac{1 - \sqrt{3}}{2} \) | Use difference of squares to simplify denominator |
| Simplify After Rationalising | Expand and simplify numerator and denominator | Check signs and combine like terms | Avoid errors that lose marks |
Active recall
Quick check
1. Simplify \( \sqrt{50} \).
\( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \)
2. Rationalise the denominator of \( \frac{3}{\sqrt{5}} \).
Multiply numerator and denominator by \( \sqrt{5} \): \( \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \)
3. Simplify \( \sqrt{81 - 16} \).
Calculate inside root: 81 - 16 = 65, so \( \sqrt{65} \) (cannot be simplified further)
4. Rationalise \( \frac{1}{2 - \sqrt{3}} \).
Multiply numerator and denominator by conjugate \( 2 + \sqrt{3} \): \( \frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3} \)