Number
Surds (Higher)
Year 10 · Year 11
- ✓By the end of this lesson students will be able to identify and define surds.
- ✓By the end of this lesson students will be able to simplify surds by extracting square factors.
- ✓By the end of this lesson students will be able to perform multiplication and division with surds.
- ✓By the end of this lesson students will be able to rationalise the denominator of a fraction involving surds, including those with binomial denominators.
Key concepts
A surd is an irrational number that can be expressed as the root of an integer. This means it cannot be written as a simple fraction a/b, where a and b are integers and b is not zero. For example, √2, √3, √5 are surds, but √4 (which is 2) and √9 (which is 3) are not, as they are rational numbers. Surds are typically square roots, but can also be cube roots or higher roots.
To simplify a surd, we look for the largest perfect square factor of the number under the root sign. We then use the rule that √(ab) = √a × √b to separate the perfect square and simplify it. The aim is to make the number under the root sign as small as possible.
Surds can be multiplied and divided using similar rules to those for simplifying. When multiplying, multiply the numbers outside the surd and the numbers inside the surd separately. When dividing, divide the numbers outside and inside the surd separately. Remember to simplify the resulting surd if possible.
Rationalising the denominator means removing any surds from the denominator of a fraction. If the denominator is a single surd, such as √a, we multiply both the numerator and the denominator by √a. This works because √a × √a = a, which is a rational number.
If the denominator is a binomial expression involving a surd, such as (a + √b) or (√a + √b), we multiply both the numerator and the denominator by its conjugate. The conjugate is formed by changing the sign between the terms (e.g., the conjugate of (a + √b) is (a - √b)). This uses the difference of two squares identity: (x + y)(x - y) = x² - y², which eliminates the surd from the denominator.
Key facts to remember
- 1A surd is an irrational root of an integer, e.g., √2, √7.
- 2The product rule for surds: √(ab) = √a × √b.
- 3The quotient rule for surds: √(a/b) = √a ÷ √b.
- 4To simplify a surd, find the largest perfect square factor and extract its root, e.g., √50 = √(25 × 2) = 5√2.
- 5To rationalise a denominator with a single surd √a, multiply numerator and denominator by √a.
- 6To rationalise a denominator with a binomial surd (a + √b), multiply numerator and denominator by its conjugate (a - √b).
- 7The conjugate pair (x + y)(x - y) always results in x² - y², which eliminates the surd if y is a surd.
Worked examples
Example 1
Simplify √72.
Answer
6√2
Always look for the largest square factor to simplify in one step. If you use a smaller factor (e.g., 4), you might need to simplify again.
Example 2
Rationalise the denominator of 10/√5.
Answer
2√5
Remember to simplify the final fraction if possible, after rationalising.
Example 3
Rationalise the denominator of 14 / (3 - √2).
Answer
6 + 2√2
Be careful with distributing the multiplication in the numerator and applying the difference of two squares correctly in the denominator.
Common mistakes
- ✗Incorrectly simplifying √(a + b) as √a + √b. This is incorrect; √(a + b) ≠ √a + √b.
- ✗Not simplifying a surd fully, for example, leaving √8 as 2√2 instead of √(4 × 2) = 2√2.
- ✗Errors when multiplying out binomials, especially when using conjugates, leading to incorrect rationalisation.
- ✗Forgetting to multiply the numerator by the same term used to rationalise the denominator.
- ✗Making arithmetic errors when squaring numbers or surds, e.g., (√3)² = 3, not √9 (which is 3, but the intermediate step can cause confusion).
Exam tips
- ★Always look for the largest square factor when simplifying surds to avoid multiple steps.
- ★When rationalising binomial denominators, clearly write out the conjugate and use brackets to ensure correct multiplication of both numerator and denominator.
- ★Show all your working steps clearly, especially when dealing with multiple operations, to gain partial marks even if the final answer has a small error.
- ★After simplifying or rationalising, check if the resulting fraction or surd can be further simplified.
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