DEPARTMENT OF MATHEMATICS
Core 1
Topic 1 and Topic 2 – Surds and Indices
1
Surds
Multiplication and Division
of Surds
Simplifying Surds
Rationalising the
denominator
Write a single surd in
simplest form
Rationalise the
denominator with a single
surd
Multiply surds with
integers
Write an expression with
multiple surds in its
simplest form
Rationalise the
denominator with a surd
of the form a ± √b
Expand Brackets with
surds
Simplify a fractional surd
with an integer
denominator
Rationalise the
denominator with a surd
of the form √a ± √b
Multiply two surds
together
Divide two surds
Simplify expressions
containing a mix of the
above
2
Surds
A little introduction:

The Natural Numbers, represented by the letter ℕ, are the positive whole numbers {1, 2, 3, 4, …}.
They are sometimes called the ‘counting numbers’.

The Integers, ℤ, include all of the whole numbers {…, -3, -2, -1, 0, 1, 2, 3, 4, …}

𝑝
The Rational Numbers, ℚ, are all numbers that can be written as ⁄𝑞 , where 𝑝 and 𝑞 are integers and 𝑞 ≠
0. Basically – all positive and negative fractions with integer numerators and denominators!

The Irrational Numbers (they don’t have a letter!) are numbers that cannot be written as a fraction in the
form above. Basically – numbers that go on forever with no repeats or patterns (think 𝜋, 𝑒, √2, etc)

The Real Numbers, ℝ, are all numbers on a number line (i.e. every number you can possibly think of!)

The Complex Numbers, ℂ, are not studied in A Level Maths, but they are pretty cool nonetheless so we’ll
briefly look at them. They involve something called the imaginary number, 𝑖, that is equal to √−1.
Surds are roots that cannot be expressed as rational numbers.
If you type a surd into your calculator and press ‘equals’, then the calculator will
round that surd to around 9 decimal places. In reality, that number continues for
ever, and without any repeats. You should always write a surd in its exact form
(i.e. as a √ ), unless you are asked to round your answer.
Rules for manipulating surds:
√𝑎 × √𝑏 = √𝑎𝑏
√𝑎
√𝑏
=√
𝑎
𝑏
√𝑎 × √𝑎 = √𝑎2 = 𝑎
3
We can simplify surds using these rules.
Worked Examples:
a)
√20
b)
√72
Simplify:
4
c)
√28
2
d)
√150 − √96 + √24
5
Notes:
6
Rationalising the Denominator:
Rationalising the denominator is the process of removing surds from the
denominator of a fraction. There are two different types of question that we will
be expected to rationalise. Let’s study the first type:
Worked Examples:
a)
b)
Rationalise:
1
√3
5√2
2√6
7
And now, let’s study the second type of question that we can be expected to
rationalise:
Worked Examples:
a)
b)
Rationalise:
1
2+√3
2
1−√5
8
c)
√5
√3+√2
9
Notes:
10
Simplifying an expression containing surds:
We can combine these different skills to simplify an expression that contains
surds. Let’s look at some examples of this:
Worked Examples:
a)
5+√2
b)
√80 +
Simplify fully the following expressions:
√3−1
30
√5
11
c)
(1+√3)(2−√2)
√6
12
Surds – Questions
Exercise 1:
Exercise 2:
13
Exercise 3:
14
Solutions to all exercises:
Exercise 1:
Exercise 2:
15
Exercise 3:
16
Indices
Basic simplification of indices
Simplifying fractional
Indices
Apply the rule:
am x an = am+n
Apply the rule:
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
Simplification of
negative indices
Apply the rule:
1
𝑎𝑚
=
𝑚
𝑎
Apply the rule:
𝑚
𝑎𝑛
=
𝑛
𝑎𝑚
Advanced Indices
Apply the rule:
1
𝑎−𝑚 = 𝑚
𝑎
Simplifying algebraic
expressions involving
indices
Apply the rule:
𝑚
𝑎𝑛
−
𝑛
𝑎
= 𝑚
𝑎
Solve an equation by changing
bases
Apply the rule:
𝑎𝑚 𝑛 = 𝑎𝑚𝑛
17
Indices
Key terms:
𝑥𝑛 →
x is the base
n is the index/power
When we are working with numbers that have the same base, we can apply the
following rules to multiply and divide the numbers:
Multiplication:
𝑎 𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
‘When multiplying, add the indices’
Division:
𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛
‘When dividing, subtract the indices’
Raising a power to a power:
(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
‘When raising to a power, multiply the indices’
Worked Examples:
Simplify:
a)
𝑥5 × 𝑥2
e)
(𝑎4 )2 × 3𝑎
b)
5𝑟 3 × 3𝑟 × 𝑟 4
f)
(2𝑎2 )3 ÷ 12𝑎5
c)
d)
𝑏7
𝑏3
9𝑏 5 ÷ 3𝑏 −2
1
2
g)
𝑟 × 3𝑟
h)
23 × 42
1
3
18
Negative and Fractional Indices:
Negative Indices:
𝑎−𝑚 =
1
𝑎𝑚
and
𝑎 −𝑚
𝑏 𝑚
( )
=( )
𝑏
𝑎
‘The ‘flipped’ number is called the reciprocal’
Fractional Indices (Basic):
1
𝑎𝑛
𝑛
= √𝑎
‘In a fractional index, the denominator is the root’
Fractional Indices (Advanced):
𝑚
𝑎𝑛
𝑛
= √𝑎𝑚
‘In a fractional index, the denominator is the root,
and the numerator is the power. The order in
which we do these doesn’t matter – pick the
easiest!’
19
Worked Examples:
a)
Evaluate the following:
5−2
2
−3
b)
( )
c)
49
1
2
d)
1
3
3
8
20
3
2
e)
25
f)
49−2
1
1
g)
h)
49
( )
2
81
(
8
)
−
2
3
125
21
Notes:
22
Converting an expression into the form 𝒌𝒙𝒏 :
Later on in C1, we will study both differentiation and integration. In these topics,
it is crucial that we can convert an expression into the form 𝑘𝑥 𝑛 .
Worked Examples:
a)
b)
Write in the form 𝑘𝑥 𝑛 :
1
𝑥3
3√𝑥
2
23
c)
d)
2
3
5 √𝑥
1
2𝑥 4
24
Notes:
25
Simplifying more complex expressions:
We also need to be able to simplify expressions with more than one letter. The
rules studied so far still apply:
Worked Examples:
Simplify:
a)
5𝑎2 𝑏 3 × 2𝑎𝑏 4
b)
35𝑎4 𝑏𝑐 3 ÷ −7𝑎2 𝑏𝑐 5
c)
√9𝑥 4
(3𝑥 2 )2
26
Notes:
27
Solving equations with indices:
Finally, we are expected to be able to solve simple equations involving indices.
It is important to check that the bases are the same when you wish to apply the
rules in an equation with multiple terms!
Worked Examples:
a)
𝑥 1/2 = 7
b)
3𝑥 3 = 81
Solve:
3
2
c)
𝑥 =8
d)
32𝑥+1 = 92𝑥−1
28
Notes:
29
Indices – Questions
Exercise 1:
30
Exercise 2:
Exercise 3:
31
Exercise 4:
Exercise 5: (Calculator Allowed)
32
Mixed Exercises: Surds and Indices
33
34
Extension Material:
35
36
37
Solutions to all exercises:
Exercise 1:
Exercise 2:
38
Exercise 3:
Exercise 4:
Exercise 5:
39
Mixed Exercise:
40
Extension Material:
41