Surds and conjugate denominators
| English | 中文 | Pinyin |
|---|---|---|
| conjugate/ˈkɒndʒuːɡeɪt/ | 共轭式 | gòng è shì |
An exact length is 1/(√5−2). Can we remove the radical from its denominator without rounding?
- An exact length is 1/(√5−2). Can we remove the radical from its denominator without rounding?
- This lesson studies conjugate 共轭式: A paired expression with the sign between two terms reversed.
Choose the mathematical structure
- For nonnegative a and b, √a×√b=√(ab), but √(a+b) is generally not √a+√b. Extract square factors to simplify surds. For a two-term denominator, multiply numerator and denominator by its conjugate: (a+b√c)(a−b√c)=a²−b²c. The original denominator and the conjugate multiplier must both be nonzero.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines conjugate?
A paired expression with the sign between two terms reversed.
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.
√72−√8=6√2−2√2=4√2. For 1/(√5−2), multiply by (√5+2)/(√5+2): the denominator is 5−4=1, giving √5+2. For 3/(2+√3), multiply by 2−√3 to get 3(2−√3)/(4−3)=6−3√3. Recombine with the original denominator to verify the exact result.
Surds and conjugate denominators
For nonnegative a and b, √a×√b=√(ab), but √(a+b) is generally not √a+√b
Check the conditions behind each index or surd manipulation.
Find the coefficient of √2 in √72−√8.
Extract squares: 6√2−2√2=4√2.
Test a tempting shortcut
- The conjugate changes one sign, not every sign. Multiply the numerator too. Squaring a sum produces a cross term: (2+√3)²=7+4√3. For real x, √(x²)=|x|, so √((−3)²)=3.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
For every real x, √(x²)=x. This claim is false. Explain which definition or assumption it violates.
Find the coefficient of √3 in 3/(2+√3) after rationalising.
Multiply by 2−√3; the denominator is 1 and the numerator is 6−3√3.
For every real x, √(x²)=x.
The conjugate changes one sign, not every sign. Multiply the numerator too. Squaring a sum produces a cross term: (2+√3)²=7+4√3. For real x, √(x²)=|x|, so √((−3)²)=3.
Interpret a new situation
- Keep exact surd expressions until a decimal is requested. Combine like surds only after simplification: √18+√8=5√2, but √2+√3 cannot be combined into one simple surd. If the conjugate is zero, this multiplier is invalid: simplify the original denominator directly, then check whether that denominator is zero. For example 1/(2+sqrt(4))=1/4 is valid even though its proposed conjugate is zero.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Evaluate √((−3)²).
The principal square root of 9 is 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
- 7357 · A-level · B. 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.
A paired expression with the sign between two terms reversed. Choose the relationship, show the method, check its assumptions and interpret the result.