1. Indices – Learning Outcomes
Use and apply rules for indices:
𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝+𝑞
𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞
𝑎 𝑝 𝑞 = 𝑎 𝑝𝑞
Use the notation 𝑎 1 2
Express rational numbers ≥ 1 in the form 𝑎× 10 𝑛 , where 𝑎 is a decimal and 𝑛 is a natural number. (Use Scientific notation).

2. Use and Apply Rules for Indices
Recall that multiplication represents repeated addition.
e.g =5×2
Indices are used to show repeated multiplication:
e.g. 2×2×2×2×2= 2 5
e.g. write the following numbers using indices:
2 is the “base”
5 is the “power” or “exponent”
3×3×3×3
4×4×4×4×4×4
8×8×8
1×1×1×1 5

3. Use and Apply Rules for Indices
e.g. Write out the following indices as numbers multiplied by themselves:
2 3
7 2
8 5
9 4
4 1
5 10

4. Use and Apply Rules for Indices
Recall that adding and subtracting are opposites, and that multiplying and dividing are opposites.
The opposite of an index is a root.
e.g is “seven squared”, 7 is “the square root of seven”.
e.g. Use a calculator to find the values of the following: 4 9 36
10000
5 2
1 1 2

5. Use and Apply Rules for Indices
What happens when numbers in index form are multiplied?
e.g. What is 2 5 × 2 3 in index form?
e.g. What is 3 2 × 3 4 in index form?
In general, what do you do to the power when multiplying numbers in index form?
𝑎 𝑝 × 𝑎 𝑞 = 𝑎 𝑝+𝑞

6. Use and Apply Rules for Indices
𝑎 𝑝 × 𝑎 𝑞 = 𝑎 𝑝+𝑞
Write the following in index form using the rule for adding powers:
3 5 × 3 3
7 2 × 7 8
2 3 × 2 1
6 2 × 6 8 × 6 3
4 1 × 4 9 × 4 2
𝑦 4 × 𝑦 2
𝑧 3 × 𝑧 10 × 𝑧 3

7. Use and Apply Rules for Indices
2012 OL P1 Q3
The table shows the values when 2 is raised to certain powers. Complete the table.
Maria wins a lottery and is given two options:
Option A: €1000 today
Option B: €2 today, €4 tomorrow, €8 the next day, doubling every day for 9 days.
Power of 2
Expanded Power of 2
Answer
2 1 2
2 2
2×2 4
2 3
2×2×2
2 4
2 5
2 6
2 7
2 8
2 9
Which is the better option?

8. Use and Apply Rules for Indices
What happens when numbers in index form are divided?
e.g. What is in index form?
e.g. What is in index form?
In general, what do you do to the power when dividing numbers in index form?
𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞

9. Use and Apply Rules for Indices
𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞
Write the following in index form using the rule for subtracting powers:
𝑦 4 𝑦 2
𝑧 3 𝑧 10
2 5 ×
6 2 ×
3 4 × × 3 6
5 2 × × 5 2
𝑦 3 × 𝑦 8 𝑦 5
𝑧 4 × 𝑧 2 𝑧 9 × 𝑧 3 × 𝑧 4

10. Use and Apply Rules for Indices
2004 OL P1 Q2
Simplify 𝑎 7 × 𝑎 4 𝑎 3 × 𝑎 2 , giving your answer in the form 𝑎 𝑛 , where 𝑛∈ℕ.
2005 OL P1 Q2
Simplify 𝑎 9 × 𝑎 5 𝑎 6 × 𝑎 2 , giving your answer in the form 𝑎 𝑛 , where 𝑛∈ℕ.

11. Use and Apply Rules for Indices
What happens when numbers in index form are raised to a power?
e.g. What is in index form?
e.g. What is in index form?
In general, what do you do to the power when raising a number in index form to a power?
𝑎 𝑝 𝑞 = 𝑎 𝑝×𝑞

12. Use and Apply Rules for Indices
𝑎 𝑝 𝑞 = 𝑎 𝑝×𝑞
Write the following in index form using the rule for multiplying powers:
𝑦 4 2 𝑧

13. Use and Apply Rules for Indices
2006 OL P1 Q2
Using a calculator or otherwise, find the exact value of
2011 OL P1 Q2
Write 𝑎 in the form 𝑎 𝑛 , 𝑛∈ℕ.
Using your answer from (i) or otherwise, evaluate
2011(S) LC OL P1 Q2
Show that 𝑎 𝑎 𝑎 4 simplifies to 𝑎

14. Use Scientific Notation
Scientific notation splits up the size of a number from its digits. It is used to show very large and very small numbers.
e.g is a very long number.
Scientific notation takes its first few digits (123…) and makes it a decimal (1.23)
It also takes its size (16 digits long) and makes it a power of ten ( ) – the power is 1 less than the number of digits.
So is written 1.23× in scientific notation, which is shorter.

15. Use Scientific Notation
To enter scientific notation mode on a CASIO, press [SHIFT], [SET UP], [7:Sci], [3] (the last one is how many digits you want).
To leave scientific notation mode on a CASIO, press [SHIFT], [SET UP], [8:Norm], [1].
e.g. enter scientific notation mode on your calculator and write the following numbers in scientific notation:
300
53 000
0.2 2

16. Use Scientific Notation

17. Use Scientific Notation
2006 OL P1 Q2
Using a calculator or otherwise, multiply 65.5 by 40 and express your answer in the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ.
2007 OL P1 Q2
Using a calculator or otherwise, multiply 54.5 by 60 and express your answer in the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ.
2010 OL P1 Q2
Using a calculator or otherwise, divide 1120 by and express your answer in the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ.
2011(S) LC OL P1 Q2
Express the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ.