What a surd is
A surd is a square root that is not a whole number: √2, √5, √12. √9 = 3 is not a surd. Surds let you give exact answers: √2 = 1.41421356... never ends, but √2 is exact.
The key rule:
√(ab) = √a × √b
Take out the biggest square factor
- √48: the largest square factor of 48 is 16
- √48 = √16 × √3
- = 4√3
Look for 4, 9, 16, 25, 36, 49, 64, 81, 100 as factors. If you use a smaller one first (√48 = 2√12), keep going: 2√12 = 2 × 2√3 = 4√3.
Like surds add like letters
4√3 + 2√3 = 6√3, just as 4x + 2x = 6x. But √3 + √5 cannot be simplified, and √12 + √27 must be simplified first: 2√3 + 3√3 = 5√3.
√a + √b is never √(a + b): √9 + √16 = 3 + 4 = 7, but √25 = 5.
Check a simplification numerically
On the calculator, work out √48 and 4√3. They should give the same decimal. Then check √12 + √27 against 5√3.