Surds

A surd is a root that isn't a whole number, such as √2 or √5. Leaving it as a surd keeps the answer exact: √2 = 1.414… goes on forever.

Learn these rules:

To simplify a surd:

1. Find the biggest square number that divides the number under the root.
2. Split the root into √(square) × √(the rest).
3. Work out the square root.

Example: simplify √72.

1. The biggest square factor of 72 is 36.
2. √72 = √36 × √2
3. 6√2

You can only add or subtract surds with the same number under the root, like collecting like terms. Simplify first to find them.

Example: simplify √50 + √8.

1. √50 = √25 × √2 = 5√2
2. √8 = √4 × √2 = 2√2
3. 5√2 + 2√2 = 7√2

To multiply, multiply the numbers in front, then the surds.

Example: simplify 2√3 × 5√6.

1. 2 × 5 = 10
2. √3 × √6 = √18 = √9 × √2 = 3√2
3. 10 × 3√2 = 30√2

Expand as you would with letters, then use √a × √a = a and collect like terms.

Example: expand (3 + √2)(1 − √2).

1. First: 3 × 1 = 3
2. Outside: 3 × −√2 = −3√2
3. Inside: √2 × 1 = √2
4. Last: √2 × −√2 = −2
5. 3 − 3√2 + √2 − 2 = 1 − 2√2

Example: expand (√5 + 2)².

1. (√5 + 2)(√5 + 2)
2. 5 + 2√5 + 2√5 + 4
3. 9 + 4√5

To rationalise a denominator is to rewrite a fraction so there's no surd on the bottom.

For a single surd, multiply the top and bottom by that surd.

Example: rationalise 6/√3.

1. Multiply top and bottom by √3: (6 × √3)/(√3 × √3)
2. = 6√3/3
3. 2√3

For a bottom like a + √b, multiply the top and bottom by a − √b. The surds cancel, because (a + √b)(a − √b) = a² − b.

Example: rationalise 4/(3 + √5).

1. Multiply top and bottom by 3 − √5
2. Bottom: (3 + √5)(3 − √5) = 9 − 5 = 4
3. Top: 4(3 − √5)
4. 4(3 − √5)/4 = 3 − √5