1. Indices – Learning Outcomes
Solve problems using the rules for indices:
𝑎 𝑝 𝑎 𝑞
𝑎 𝑝 𝑎 𝑞
𝑎 0
𝑎 𝑝 𝑞
𝑎 1 𝑞
𝑎 𝑝 𝑞
𝑎 −𝑝
𝑎𝑏 𝑝
𝑎 𝑏 𝑝

2. Solve Problems Using 𝑎 𝑝 𝑎 𝑞
Recall the meaning of indices:
𝑎 𝑝 =𝑎×𝑎×𝑎×𝑎×𝑎×…
e.g. 𝑎 4 =𝑎×𝑎×𝑎×𝑎
Write 𝑎 4 𝑎 2 as a product of 𝑎s. Then, write it as a single power of 𝑎.
Write 𝑏 3 𝑏 6 as a product of 𝑏s. Then, write it as a single power of 𝑏.
Write a general rule for combining powers when given 𝑎 𝑝 𝑎 𝑞 .

3. Solve Problems Using 𝑎 𝑝 𝑎 𝑞
Simplify each of the following:
3 7 × 3 8
5 2 × 5 3 × 5 5
7 8 × 8 7 × 8 3 × 7 2
𝑎 3 × 𝑎 10
𝑏 10 × 𝑏 −2
𝑐 4 × 𝑐 −2 × 𝑑 10 × 𝑑 3
Solve for 𝑥:
4 𝑥 × 4 2 =1024
3 𝑥+1 ×3=81

4. Solve Problems Using 𝑎 𝑝 𝑎 𝑞
Write 𝑎 4 𝑎 2 as a product of 𝑎s. Then, write it as a single power of 𝑎.
Write 𝑏 6 𝑏 3 as a product of 𝑏s. Then, write it as a single power of 𝑏.
Write a general rule for combining powers when given 𝑎 𝑝 𝑎 𝑞 .

5. Solve Problems Using 𝑎 𝑝 𝑎 𝑞
Simplify each of the following:
6 2 ×
7 8 × × 7 2
𝑎 7 𝑎 2
𝑏 10 𝑏 −2
𝑐 4 × 𝑐 −2 𝑑 10 × 𝑑 3
Solve for 𝑥:
4 𝑥 =64
3 3𝑥 𝑥 =27

6. Solve Problems Using 𝑎 0
Combine the powers of 𝑎 in 𝑎 5 𝑎 5 using the 𝑎 𝑝 𝑎 𝑞 rule.
Write 𝑎 5 𝑎 5 as a product of 𝑎s.
Compare the results of parts 1 and 2.
Combine the powers of 𝑏 in 𝑏 3 𝑏 3 using the 𝑏 𝑝 𝑏 𝑞 rule.
Write 𝑏 3 𝑏 3 as a product of 𝑏s.
Compare the results of parts 4 and 5.
Generalise the value of 𝑎 0 .

7. Solve Problems Using 𝑎 𝑝 𝑞
Write 𝑎 as a product of 𝑎s. Then, write it as a single power of 𝑎.
Write 𝑏 as a product of 𝑏s. Then, write it as a single power of 𝑏.
Write a general rule for combining powers when given 𝑎 𝑝 𝑞 .

8. Solve Problems Using 𝑎 0 , 𝑎 𝑝 𝑞
Simplify each of the following:
𝑎 5 2
𝑏 7 0 𝑐
Solve for 𝑥:
5 2 𝑥 = 5 10
81 𝑥−3 =1

9. Solve Problems Using 𝑎 1 𝑞
Use the 𝑎 𝑝 𝑞 rule to evaluate:
, then find
, then find
Generalise the meaning of 𝑎
, then find
Generalise the meaning of 𝑎 1 𝑞 .

10. Solve Problems Using 𝑎 𝑝 𝑞
Given the previous rules:
𝑎 1 𝑞 = 𝑞 𝑎
𝑎 𝑝 𝑞 = 𝑎 𝑝×𝑞
Generalise the rule for 𝑎 𝑝 𝑞

11. Solve Problems Using 𝑎 1 𝑞 and 𝑎 𝑝 𝑞
Simplify each of the following:
9 1 2
3 4 2
𝑏 5 2
𝑎 5 × 𝑎 3 2
3 𝑎 9
Solve for 𝑥:
4 𝑥+2 = 1024
5 𝑥−3 = 𝑥

12. Solve Problems Using 𝑎 −𝑝
Given the previous rules:
𝑎 0 =1
𝑎 𝑝−𝑞 = 𝑎 𝑝 𝑎 𝑞
Generalise the rule for 𝑎 −𝑝

13. Solve Problems Using 𝑎𝑏 𝑝
Write 3×5 4 as a product of 3s and 5s. Then, write it as a product of 3 to a single power and 5 to a single power.
Write 𝑎𝑏 3 as a product of 𝑎s and 𝑏s. Then, write it as a product of 𝑎 to a single power and 𝑏 to a single power.
Write a general rule for combining powers when given 𝑎𝑏 𝑝 .

14. Solve Problems Using 𝑎 𝑏 𝑝
Write as a quotient of 3s and 5s. Then, write it as a quotient of 3 to a single power and 5 to a single power.
Write 𝑎 𝑏 3 as a quotient of 𝑎s and 𝑏s. Then, write it as a quotient of 𝑎 to a single power and 𝑏 to a single power.
Write a general rule for combining powers when given 𝑎 𝑏 𝑝 .

15. Solve Problems Using the Rules for Indices
Solve for 𝑥:
4 𝑥 −5 2 𝑥 =0
5 2 𝑥 − 4 𝑥 =4
2 𝑒 2𝑥 −3 𝑒 𝑥 −2=0
𝑎 𝑥 = 1 𝑎 𝑥
Given 𝑓 𝑛 = 𝑛 𝑛 . Write 𝑓 𝑛+1 −𝑓(𝑛) in the form 𝑎 2 4𝑛 +𝑏( 3 4𝑛 ).