How do you rationalise the denominator?

Defintions

Rationalising the denominator means rewriting a fraction so that the bottom only contains rational numbers.

Remember: A rational number is one that can be made by dividing two whole numbers (e.g. 3 or 1/2)

Rules for how to do

1: If 1/ √a then multitply top and bottom by √ a

2: If 1/ (a+√ b) then multiply top and bottom by (a-√ b)

3: If 1/ (a-√ b) then mulitply top and bottom by (a+√ b)

4: If 1/ (√b + √c) then mulitply top and bottom by (√b - √c)

Note: When we multiply with something where the top and bottom are the same, it is just like we are mulitplying by 1

Examples

1: "Rationalise the denominator for 5/ √ 3"

5/ √ 3 = (5/ √ 3) x (√ 3 / √ 3)

         = (5√ 3) / 3

2: "Rationalise the denominator for 1/ (3+√ 2)"

1/ (3+√ 2) = 1/ (3+√ 2) x [(3 - √2) / (3 - √2)]

                 = (3- √2) / (9-2)

                 = (3- √2) / 7

3: "Rationalise the denominator for (√5+√2)/(√5 -√2)"

(√5+√2)/(√5 -√2)=(√5+√2)/(√5 -√2) x (√5+√2)/(√5 +√2)

                          = (5 + 2√10 +2) / (5-2)

                          = (7+ 2√10) / 3