A Level Mathematics for AQA Student Book 1 One important method used when dealing with surds in fractions is called rationalising the denominator. This technique removes the surd from the denominator. WORKED EXAMPLE 2.10 Rationalise the denominator of 2 . 3 2 2× 3 = 3 3× 3 2 3 = 3
Multiply top and bottom by 3 .
You can use the difference of two squares to rationalise the denominator of more complicated expressions. a 2 − b 2 = (a − b)(a + b). So for an expression such as 5 + 3 , multiplying it by 5 − 3 gives
52 − ( 3 ) = 22. Importantly, this product is rational. 2
Gateway to A Level See Gateway to A Level Section B for revision of the difference of two squares factorisation.
Key point 2.3 To rationalise the denominator of a fraction, multiply top and bottom by the appropriate expression to give the difference of two squares.
WORKED EXAMPLE 2.11 Rationalise the denominator of
3 . 8−2 3
3× (8 +2 3 ) 3 = 8 −2 3 (8 −2 3 )× (8 +2 3 ) =
24 + 6 3 64 − 4 × 3
= 24 + 6 3 52 12 + 3 3 = 26
20
By inspection, the appropriate term to multiply top and bottom by is 8 + 2 3 . You don’t need to multiply the bottom out – use the identity for the difference of two squares. There is a common factor of 2 on top and bottom so cancel it.