1. What does this symbol, ⇔, mean?
How do you find the converse of a theorem?
Which numbers do you need to test in order to prove by exhaustion that 37 is a prime?

2. Is implied by and implies, if and only if, is equivalent to
To find the converse, change over the If and the then, and write the statement so that it reads sensibly.
1, 2, 3, 4, 5, 6

3. Chapter 2.2 Working with Indices
“It’s not that I’m so smart, it’s just that I stay with problems longer.”
- Albert Einstein

4. 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

5. 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.

6. 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

7. 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

8. 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 =

9. 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

10. 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

11. Let’s see if we can put this into practice!

12. Odd one out activity

13. Put these in ascending order

15. Have you played Blockbusters before?

16. Countdown You don’t have to use all numbers
You can only use each number once
You can only use indices and multiplication

20. Quick Standard form activity

21. …and a standard form activity to make you think!

22. Right lets finish the week with some real brain teasers
You may want to work together one some of these and share ideas