What surds are (Higher)
- A surd is a square root (or other root) that cannot be written as a whole number or a fraction, such as √2 and √3.
- Surds are exact values. Leaving an answer as a surd is more accurate than rounding a decimal.
Simplifying surds (Higher)
- Use √(ab) = √a × √b, looking for a square number factor: √12 = √4 × √3 = 2√3 and √50 = √25 × √2 = 5√2.
- √a × √a = a, so √3 × √3 = 3 and (√5)² = 5.
- √a × √b = √(ab): √2 × √8 = √16 = 4.
Adding and expanding (Higher)
- Only like surds can be added: 3√5 + 2√5 = 5√5. Simplify first: √18 + √8 = 3√2 + 2√2 = 5√2.
- Expand brackets as normal: (2 + √3)² = 4 + 2√3 + 2√3 + 3 = 7 + 4√3.
- (a + √b)(a − √b) = a² − b, with no surd left: (1 + √2)(1 − √2) = 1 − 2 = −1.
Rationalising the denominator (Higher)
- To remove a surd from the denominator, multiply the top and bottom by that surd: 1/√3 = √3/3 and 6/√2 = 6√2/2 = 3√2.
- If the denominator is like 3 − √5, multiply the top and bottom by 3 + √5: 4/(3 − √5) = 4(3 + √5)/(9 − 5) = 3 + √5.
Key terms
- Surd
- A root that can't be written as a whole number or fraction, such as √2.
- Simplify a surd
- Write it with the smallest possible number under the root, such as √12 = 2√3.
- Like surds
- Surds with the same number under the root, which can be added.
- Rationalise the denominator
- Rewrite a fraction so there is no surd on the bottom.
- Exact value
- An answer given without rounding, such as 5√2.