Exact surds and rationalising denominators · Higher
| English | 中文 | Pinyin |
|---|---|---|
| surd/sɜːd/ | 根式 | gēn shì |
Can a root stay exact?
- A square has area 12 cm². Can its side be written exactly without a long calculator decimal?
- This lesson studies surd 根式: An irrational root kept in exact form rather than replaced by a rounded decimal.
Choose the mathematical structure
- Extract square factors: √(a²b)=a√b for a≥0,b≥0. Add only matching root parts. Multiply roots with nonnegative radicands. To rationalise a denominator, multiply numerator and denominator by the same suitable root or conjugate.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines surd?
An irrational root kept in exact form rather than replaced by a rounded decimal.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The side is √12=√(4×3)=2√3 cm. Thus √12+√27=2√3+3√3=5√3. Also 6/√3=6√3/3=2√3. For 1/(2+√3), multiply by (2-√3)/(2-√3): the denominator becomes 4-3=1, giving 2-√3. A circle of radius 3 has exact area 9π; a decimal is an approximation.
Exact surds and rationalising denominators
Extract square factors: √(a²b)=a√b for a≥0,b≥0
Compare the model with the worked case and explain one change.
Find the coefficient of √3 in √12.
12=4×3, so √12=2√3.
Test a tempting shortcut
- √(a+b) generally differs from √a+√b. Match the radicand before collecting terms. Rationalising changes the form, not the value; multiplying only the denominator changes the value.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
For every pair of positive numbers, √(a+b)=√a+√b. This claim is false. Explain which definition or assumption it violates.
Find the coefficient of √3 in √12+√27.
√12=2√3 and √27=3√3; add coefficients 2+3=5.
For every pair of positive numbers, √(a+b)=√a+√b.
√(a+b) generally differs from √a+√b. Match the radicand before collecting terms. Rationalising changes the form, not the value; multiplying only the denominator changes the value.
Interpret a new situation
- AQA N8 Higher requires exact surds and rationalisation. Foundation retains exact fractions and multiples of π; do not assign this surd lesson to Foundation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the coefficient of √3 in 6/√3.
Multiply by √3/√3: 6√3/3=2√3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An irrational root kept in exact form rather than replaced by a rounded decimal. Choose the relationship, show the method, check its assumptions and interpret the result.