1. Laws of Indices

2. Working with Indices
Write the following numbers as the base 5 raised to a power.
𝑖 𝑖𝑖 𝑖𝑖𝑖 (𝑖𝑣) 5 5
Chapter 2.2
Base
Index
Power
Standard Form
Notice that = 1, = 5, = 25, = 125, = 625,
𝑖𝑖𝑖 = = 5 −3
𝑖 = 5 4
𝑖𝑣 = 5 1 × =
𝑖𝑖 1 = 5 0

3. Working with Indices Simplify these.
𝑖 𝑖𝑖 𝑖𝑖𝑖 4 − (𝑖𝑣)
Chapter 2.2
There are two ways to approach this.
Base
Index
Power
Standard Form
𝑖 = = 4096
𝑎 = = =4
𝑖𝑖 = 3 27 =3
𝑏 = = =4
𝑖𝑖𝑖 − 5 2 = − 5 2 = 2 2× − = 2 −5 = 1 32
Both give the same answer, and both are correct.

4. Working with Indices Simplify (4 2 × 1 16 × 5 32 ) 2 Chapter 2.2 Base
= × × ×
( × × ) 2
4, 16 and 32 are all powers of 2
Base
Index
Power
Standard Form
You need the same base
number to add powers
= −
= −
= 2 −1
= 1 2

5. Working with Indices Solve the equation 2 𝑎 × 6 𝑏 = 48 Chapter 2.2
Start by writing each number as a product of its prime factors.
Chapter 2.2
2 𝑎 × (2 × 3) 𝑏 = × 3
Notice the use of the rule
(𝑥 × 𝑦) 𝑛 = 𝑥 𝑛 × 𝑦 𝑛 ,
with 𝑥 = 2, 𝑦 = 3 and 𝑛 = 𝑏.
Base
Index
Power
Standard Form
2 𝑎 × 2 𝑏 × 3 𝑏 = × 3
⇒ 2 𝑎+𝑏 × 3 𝑏 = × 3
Comparing powers, this gives the pair of simultaneous equations:
𝑎+𝑏 =4
𝑏=1
So 𝑎=3, 𝑏=1

6. Working with Indices Simplify 5 1 2 − 5 3 2 + 5 5 2 Chapter 2.2 Base
This can be written as:
Base
Index
Power
Standard Form
5 − = = =
= (1 − ) 5 =

7. Working with Indices Simplify 3𝑥 (𝑥 + 7) 1 2 − 2 (𝑥 + 7) 3 2
Chapter 2.2
3𝑥 (𝑥 + 7) − 2(𝑥 + 7) (𝑥 + 7) 1 2
This expression is:
(𝑥 + 7) is a common factor
Base
Index
Power
Standard Form
= 3𝑥−2 𝑥+7 (𝑥 + 7) 1 2
=(𝑥−14) (𝑥 + 7) 1 2
or (𝑥+14) 𝑥+7

8. Working with Indices
Light travels at a speed of 300 million metres per second. At a certain time Pluto is 5.4 terametres from Earth. How many hours does it take light to travel from Pluto to Earth?
Chapter 2.2
Base
Index
Power
Standard Form
= 5.4× × 10 8
Time taken
= 1.8 × seconds
= hours
There are 60 x 60 seconds in an hour
=5 hours

9. Surds

10. Using and Manipulating Surds
Simplify the following, giving your answers in the simplest surd form.
(𝑖) 32
(𝑖𝑖) 4( ) − 3( 7 − )
(𝑖𝑖𝑖) 2( 𝑥 + 𝑦 ) − 3( 𝑥 − 𝑦 )
Chapter 2.1
Surd
Simplest Form
Rationalise
Denominator

11. Using and Manipulating Surds
The largest square number
which is a factor of 32 is 16.
(𝑖) 32
= 16 × 2
= × 2
Chapter 2.1
= 4 2
Multiply out the brackets.
𝑖𝑖 4( ) − 3( 7 − )
= −
Surd
Simplest Form
Rationalise
Denominator
Collect like terms. =
(𝑖𝑖𝑖) 2( 𝑥 + 𝑦 ) − 3( 𝑥 − 𝑦 )
= 2 𝑥 𝑦 − 3 𝑥 𝑦
= − 𝑥 𝑦
It’s a good idea to write the positive term first since it is easy to lose a minus sign at the start of an expression.
= 5 𝑦 − 𝑥

12. Using and Manipulating Surds
Multiply out and simplify:
(𝑖) ( )( )
(𝑖𝑖) ( )( 5 − 3 )
Chapter 2.1
Surd
Simplest Form
Rationalise
Denominator

13. Using and Manipulating Surds
(𝑖) ( )( )
= 2 ( ) ( )
= ( 2 × 2 ) + ( 2 × ) + ( 3 × 2 ) + ( 3 × )
= (2 × 3)
Chapter 2.1
Notice that the first and last terms are rational numbers. =
Surd
Simplest Form
Rationalise
Denominator
= − 5 × × 5 −
𝑖𝑖 − 3
= 5 − − 3
= 5 – 3
= 2

14. Using and Manipulating Surds
Simplify the following, giving your answers in the simplest surd form.
(𝑖) 𝑖𝑖
(𝑖𝑖𝑖)
Chapter 2.1
Surd
Simplest Form
Rationalise
Denominator

15. Using and Manipulating Surds
= ×
(𝑖) 4 2
Multiplying top and
bottom by 2 =
Chapter 2.1
=2 2
Surd
Simplest Form
Rationalise
Denominator

16. Using and Manipulating Surds
(𝑖𝑖)
= ×
Multiplying top and
bottom by 6 =
Chapter 2.1 =
Surd
Simplest Form
Rationalise
Denominator =
This can also be written as

17. Using and Manipulating Surds
= × 3− − 5
(𝑖𝑖𝑖)
Rationalising the
denominator
= 3− −
Chapter 2.1
= 3− 5 9−5
Surd
Simplest Form
Rationalise
Denominator
= 3− 5 4

18. Using and Manipulating Surds
Simplify the following, giving your answers in the simplest surd form.
(𝑖) 𝑥 𝑦 𝑥
(𝑖𝑖) 𝑥+2 𝑦 𝑥+ 𝑦
Chapter 2.1
Surd
Simplest Form
Rationalise
Denominator

19. Using and Manipulating Surds
= 𝑥 𝑦 𝑥 × 𝑥 𝑥
(𝑖) 𝑥 𝑦 𝑥
Multiplying top and
bottom by 𝑥
= 𝑥 𝑥𝑦 𝑥
Chapter 2.1
In algebra it is usual to write
letters in alphabetical order
= 𝑥𝑦
Surd
Simplest Form
Rationalise
Denominator
= 𝑥+2 𝑦 𝑥+ 𝑦 × 𝑥− 𝑦 𝑥− 𝑦
𝑖𝑖 𝑥+2 𝑦 𝑥+ 𝑦
Rationalising the
denominator 
= 𝑥 2 −𝑥 𝑦 +2𝑥 𝑦 −2𝑦 𝑥 2 − 𝑦 2
= 𝑥 2 +𝑥 𝑦 −2𝑦 𝑥 2 −𝑦

20. Now let’s have a go at the booklet