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

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 .

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

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 .

Solve Problems Using 𝑎 𝑝 𝑎 𝑞 Simplify each of the following: 3 5 3 2 6 2 × 6 7 6 3 7 8 × 8 7 8 3 × 7 2 𝑎 7 𝑎 2 𝑏 10 𝑏 −2 𝑐 4 × 𝑐 −2 𝑑 10 × 𝑑 3 Solve for 𝑥: 4 𝑥 4 2 =64 3 3𝑥+1 3 𝑥 =27

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 .

Solve Problems Using 𝑎 𝑝 𝑞 Write 𝑎 2 5 as a product of 𝑎s. Then, write it as a single power of 𝑎. Write 𝑏 3 4 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 .

Solve Problems Using 𝑎 0 , 𝑎 𝑝 𝑞 Simplify each of the following: 5 2 3 3 0 6 2 5 4 2 3 𝑎 5 2 𝑏 7 0 𝑐 3 2 4 Solve for 𝑥: 5 2 𝑥 = 5 10 81 𝑥−3 =1

Solve Problems Using 𝑎 1 𝑞 Use the 𝑎 𝑝 𝑞 rule to evaluate: 9 1 2 2 , then find 3 2 . 25 1 2 2 , then find 5 2 . Generalise the meaning of 𝑎 1 2 . 64 1 3 3 , then find 4 3 . Generalise the meaning of 𝑎 1 𝑞 .

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

Solve Problems Using 𝑎 1 𝑞 and 𝑎 𝑝 𝑞 Simplify each of the following: 9 1 2 3 4 2 4 5 24 𝑏 5 2 𝑎 5 × 𝑎 3 2 3 𝑎 9 Solve for 𝑥: 4 𝑥+2 = 1024 5 𝑥−3 = 125 𝑥

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

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 .

Solve Problems Using 𝑎 𝑏 𝑝 Write 3 5 4 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 .

Solve Problems Using the Rules for Indices Solve for 𝑥: 4 𝑥 −5 2 𝑥+1 +16=0 5 2 𝑥 − 4 𝑥 =4 2 𝑒 2𝑥 −3 𝑒 𝑥 −2=0 𝑎 𝑥 = 1 𝑎 𝑥 Given 𝑓 𝑛 =24 2 4𝑛 + 3 4𝑛 . Write 𝑓 𝑛+1 −𝑓(𝑛) in the form 𝑎 2 4𝑛 +𝑏( 3 4𝑛 ).