Surds preserve exact information. Writing √2 instead of 1.414… is not merely a preference for symbols; it keeps an exact value available for later algebra, trigonometry and checking.
1. What is a surd?
A surd is an irrational root left in exact form, such as √2, √5 or 3√7. A root such as √9=3 is not left as a surd because it simplifies exactly to a rational number.
2. Simplify by extracting perfect-square factors
√72=√(36×2)=6√2. Likewise, √200=√(100×2)=10√2.
The aim is to leave no perfect-square factor greater than 1 inside the square root.
3. Like surds combine like algebraic terms
3√2+5√2=8√2. But √2+√3 cannot be simplified to √5. Addition does not combine radicands.
First simplify each surd. For example, √8+√18=2√2+3√2=5√2.
4. Multiplication follows root laws
For nonnegative real quantities, √a√b=√(ab). Thus √6×√24=√144=12.
With coefficients, (2√3)(5√6)=10√18=30√2.
5. Expand brackets carefully
(√3+2)(√3−1)=3−√3+2√3−2=1+√3.
Surds behave as algebraic quantities. Standard distributive rules still apply.
6. Conjugates create rational differences of squares
The conjugate of a+b√c is a−b√c. Their product is a²−b²c because the middle surd terms cancel.
For example, (3+√5)(3−√5)=9−5=4.
7. Exact values support trigonometry
sin45°=√2/2 and cos30°=√3/2. Keeping these exact values makes identities and later symbolic calculations transparent. Premature decimal conversion can obscure cancellation.
8. Exact does not mean unsimplified
√50 is exact, but 5√2 is usually a more useful simplified exact form. Similarly, 2/√3 is exact but can be rewritten with a rational denominator as 2√3/3.
9. Common mistakes
- √a+√b=√(a+b): generally false.
- Forgetting to simplify the radicand before combining terms.
- Converting to decimals before exact algebra is finished.
- Applying real square-root rules without checking nonnegative quantities.
10. Practice
- Simplify √48.
- Simplify √12+√27.
- Evaluate (2√5)(3√10).
- Expand (√7+3)(√7−2).
- Evaluate (4+√3)(4−√3).
11. Answers
- 4√3.
- 2√3+3√3=5√3.
- 6√50=30√2.
- 7−2√7+3√7−6=1+√7.
- 16−3=13.
Continue through the Additional Mathematics Algebra Guide or return to the Additional Mathematics Hub.