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?

5 12 × 5 18 = 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.

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.

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.

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

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

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?

(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.

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

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

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

2 6 ÷2 2 = 2 ? Division involving indices: 2 6 ÷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?

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 ÷ (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?

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

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.

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

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 2 𝑎 𝑝 Task: Task: All the laws of indices

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