Surds and conjugate denominators
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| conjugate/ˈkɒndʒuːɡeɪt/ | liên hợp |
Can we remove the radical from its denominator without rounding?
- An exact length is 1/(√5−2).
- This lesson studies conjugate 共轭式: A paired expression with the sign between two terms reversed.
The method
- 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.
Why is √((-3)²) equal to 3 and not -3?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
(-3)²=9 and √9=3 by the principal-root convention. Writing √(x²)=x fails for every negative x; the absolute value is required.
A worked case
√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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Extract squares: 6√2−2√2=4√2.
Where it goes wrong
- 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.
Not true: For every real x, √(x²)=x. √(x²)=|x|: for x=−3 it gives 3, not −3.
Find the coefficient of √3 in 3/(2+√3) after rationalising.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Multiply by 2−√3; the denominator is 1 and the numerator is 6−3√3.
Conjugate means a paired expression with the sign between two terms reversed.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True: A paired expression with the sign between two terms reversed.
Applying it
- 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.
Evaluate √((−3)²).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The principal square root of 9 is 3.
Match each surd operation to its result.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The conjugate flips one sign so the product loses its surd part; multiply numerator by the same conjugate to preserve the value.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Conjugate: A paired expression with the sign between two terms reversed.
- For nonnegative a and b, √a×√b=√(ab), but √(a+b) is generally not √a+√b.
- Watch for: the conjugate changes one sign, not every sign.