surds

a√b + c√b = (a + c)√b

a√b - c√b = (a - c)√b

Examples 4√7 - 2√7 = 2√7.

5√2 + 8√2 = 13√2

NB1: 5√2 + 3√3 cannot be manipulated because the surds are different (one is √2 and one is √3).

NB2: √a + √b is not the same as √(a + b) .

√ab = √a × √b
√(a/b) = √a/√b

Examples

√5 × √15 = √75
= √25 × √3
= 5√3.

(1 + √3) × (2 - √8)            [The brackets are expanded as usual]
= 2 - √8 + 2√3 - √24
= 2 - 2√2 + 2√3 - 2√6

a) 1 
  √2 .

b) 1 + 2 
 1 - √2

a) Multiply the top and bottom of the fraction by √2. The top will become √2 and the bottom will become 2 (√2 times √2 = 2).

b) In situations like this, look at the bottom of the fraction (the denominator) and change the sign (in this case change the minus into plus). Doing this forms the conjugate of the denominator. Now multiply the top and bottom of the fraction by this.

Therefore:
1 + 2  =   (1 + 2)(1 + √2)  =  1 + √2 + 2 + 2√2  =  3 + 3√2
1 - √2       (1 - √2)(1 + √2)      1 + √2 - √2 - 2         - 1

= -3(1 + √2)