CHAPTER 3. SURDS 3.2
QUESTIONSimplify:√
147 +
√
108
SOLUTIONStep 1 : Simplify each square root by converting each number to a product of it’s prime
factors√
147 +√
108 =
√
49 × 3 +
√
36 × 3
=
�
72 × 3 +
�
62 × 3
Step 2 : Square root all squarednumbers= 7
√
3 + 6
√
3
Step 3 : The exact same surds can be treated as ”like terms” and may be added= 13
√
3
See video: VMecu at http://www.everythingmaths.co.zaRationalising Denominators EMBM
It is useful to work withfractions, which have rational denominators instead of surd denominators. It is
possible to rewrite any fraction, which has a surd in the denominator asa fraction which has a rational
denominator. We will now see how this can beachieved.
Any expression of the form
√
a+√
b (where a and b are rational) can be changed into a rational number
by multiplying by
√
a−√
b (similarly√
a−√
b can be rationalised by multiplying by√
a +√
b). This
is because
(
√
a +√
b)(√
a−√
b) = a− b (3.8)which is rational (since a and b are rational).
If we have a fraction which has a denominator which looks like
√
a+√
b, then we can simply multiplythe fraction by
√a−√b
√a−√bto achieve a rational denominator. (Remember that√a−√b
√a−√b= 1)c
√
a +√
b=
√
a−√
b
√
a−√
b×
c
√
a +√
b(3.9)
=
c√
a− c√
b
a− b