1. 2 7 × 2 6 = 2 ?
2 ×2×2×2×2×2×2
2 ×2×2×2×2×2
In total how many times will we be multiplying 2’s together?

2. 5 12 × = 5 ?
5 ×5×5×5 ×5×5×5 ×5×5×5 ×5×5
5 ×5×5×5 ×5×5×5 ×5×5×5 ×5×5 ×5×5×5 ×5×5×5
Explain how you know that it will be 5^30. What is the rule here? If we are multiplying a number m times, and then multiplying it n times, in total we are multiplying m + n times.

3. 5 𝑚 × 5 𝑛 = 5 ?
5 ×5×5×5 ×5×5×5 ×5×5×5 ×5×5 …
5 ×5×5×5 ×5×5×5 ×5×5×5 ×5×5 ×5×5×5 ×5×5×5…..
Explain how you know that it will be 5^30. What is the rule here? If we are multiplying a number m times, and then multiplying it n times, in total we are multiplying m + n times.

4. 𝑎 𝑚 × 𝑎 𝑛 = 𝑎 𝑚+𝑛
The law tells us that when we are multiplying numbers that have the same base, we can add the indices.

5. 3 0 × 3 3
𝑎× 𝑎 −2
𝑎 2 𝑏× 𝑎𝑏 3
Example 2: a alone means a to the power 1. The last example cannot be simplified: the base numbers must be the same if we are to add the indices.
𝑧 2 × 𝑥 3

6. 2 0 × 2 8 2 −3 ×2 𝑎 3 × 𝑏 3 𝑎 4 × 𝑎 −1 𝑐 −4 × 𝑐 3 2𝑎 × 𝑎 3 2 𝑏 4 ×𝑎𝑐
2 0 × 2 8
2 −3 ×2
𝑎 3 × 𝑏 3
(−5) 2 × 5 4
𝑎 4 × 𝑎 −1
𝑐 −4 × 𝑐 3
2𝑎 × 𝑎 3
2 𝑏 4 ×𝑎𝑐
𝑎𝑏 × 𝑏 3
Task: Algebraic Multiplication Squares

7. simplify
(𝑎 𝑏 3 ) 2
Part 2: Can we write this out in full, what does it mean? How many a’s will I multiply? How many b’s will I multiply? Can we write it without the brackets?

8. (𝑎 𝑏 3 ) 2 𝑎 ×𝑏×𝑏×𝑏 𝑎 ×𝑏×𝑏×𝑏 simplify ×
We end up with b to the power 6. We can multiply the powers because we are getting b 3 times, twice. i.e. 3 x 2 times.

9. simplify
( 𝑎 3 𝑏 2 ) 4
Can we write this out in full, what does it mean?

10. ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚𝑛
The law tells us that when one power is raised to another power, we can multiply the powers.

11. 3 ×
𝑎 2 × ( 2𝑎 4 𝑏) 3
5 𝑞 2 × 𝑞
Write as a single power.

12. 2 6 ÷2 2 = 2 ? Division involving indices:
2 6 ÷ = 2 ?
How else can we think about what this means? We are multiplying 2 six times, then when we get our result, we are dividing by 2 twice. How many 2’s will we end up with?

13. 2 6 ÷2 2 = 2 ? 2 ×2×2×2×2 ×2 ÷ (2×2) Division involving indices:
2 6 ÷ = 2 ?
2 ×2×2×2×2 ×2
÷ (2×2)
How else can we think about what this means? We are multiplying 2 six times, then when we get our result, we are dividing by 2 twice. How many 2’s will we end up with?

14. Division involving indices:
𝑎 5 ÷𝑎 2 = 𝑎 ?
𝑎 ×𝑎×𝑎×𝑎×𝑎
(𝑎×𝑎)

15. 𝑎 𝑚 ÷𝑎 𝑛 = 𝑎 𝑚−𝑛
This law tells us that if we are dividing, we can subtract the powers. It works for negative and zero powers too.

16. 2 3 ÷ 2 −3
4𝑎 2 𝑏 ÷2𝑎 𝑏 4
12 𝑏 4 𝑐 3𝑏 𝑐 2
Write as a single power.

17. 3 2 ÷ 3 4
2 −3 ÷ 3 −1
𝑏 3 ÷ 𝑏 0
𝑏 4 𝑐 𝑏 𝑐 2
(5𝑏) 2 ÷ (5 𝑏 3 ) 2
3𝑝 2 ÷ 𝑝 2
10𝑎𝑐 5 𝑎 2 𝑐 2
(5 𝑏 3 ) 2 𝑏 −1
𝑎 𝑝 𝑎 𝑝
Task: Task: All the laws of indices

18. 𝑎 𝑚 ÷𝑎 𝑛 = 𝑎 𝑚−𝑛 𝑎 𝑚 × 𝑎 𝑛 = 𝑎 𝑚+𝑛 ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚𝑛
𝑎 𝑚 ÷𝑎 𝑛 = 𝑎 𝑚−𝑛
𝑎 𝑚 × 𝑎 𝑛 = 𝑎 𝑚+𝑛
( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚𝑛
All three laws. Task: All the laws of indices