--- title: "Index laws" subtitle: "Irrational numbers and index laws · Index laws · Lesson 5" --- # Index laws **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 2¹⁰ × 2¹⁰ has over a million as its answer, but you can write it down in a second: 2²⁰. Write the powers out once, and you will see why. :::teacher-note **In one sentence:** Students write powers out as factors on the board and discover the product, quotient and power laws and why a⁰ = 1 (AC9M8N02, node 1). **Students should be able to say:** I can explain each index law by writing the powers out. I can use the laws to simplify powers of the same base. I can explain why a⁰ = 1. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Write each power out as 2s. :::keep-together **2³ × 2⁴ = 2 to what power?** ☐ 7 ☐ 1 ☐ 12 ☐ It is 4⁷ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 2⁷: three 2s times four 2s is seven 2s multiplied together. The base stays 2. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Three laws and zero :::keep-together ### Divide Here is the routine. 1. $3^5 \div 3^2 = \frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$ — Write both powers out. 2. $= 3 \times 3 \times 3 = 3^3$ — Two pairs cancel: 5 − 2 = 3 factors left. 3. $(2^3)^2 = 2^3 \times 2^3 = 2^6$ — A power of a power: 2 groups of 3 factors, 3 × 2. 4. $2^3 \div 2^3 = 1 = 2^0$ — Everything cancels: so 2⁰ = 1. **5⁶ ÷ 5² = 5 to what power?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 Two pairs of 5s cancel, leaving four: 5⁴. ::: :::keep-together ### Find the mistake **Kai simplifies 3² × 3⁴. Click the line where his working goes wrong.** ``` 1 Kai simplifies 3² × 3⁴. 2 The indices add: 2 + 4 = 6. 3 The bases multiply: 3 × 3 = 9. 4 So 3² × 3⁴ = 9⁶. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: the base stays 3. There are six 3s multiplied together: 3⁶ (= 729), not 9⁶. ::: :::keep-together ### Zero **Why is 5⁰ equal to 1?** ☐ 5³ ÷ 5³ = 1, and the quotient law gives 5³⁻³ = 5⁰ ☐ Because any number to a power is at least 1 ☐ It is a rule with no reason ☐ Because 0 × 5 = 0, and then add 1 ::: :::teacher-note **Answer:** 5³ ÷ 5³ = 1, and the quotient law gives 5³⁻³ = 5⁰ Dividing anything by itself gives 1, and the law says it is a⁰. ::: :::typical ## Each law on the board **Quotient.** **Power of a power.** **Zero index.** **Power or product.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Index laws :::keep-together ### Product **7⁵ × 7³ = 7 to what power?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 5 + 3 = 8: 7⁸. ::: :::keep-together ### Power of a power **(4²)⁵ = 4 to what power?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 10 5 groups of 2: 2 × 5 = 10, so 4¹⁰. ::: :::keep-together ### Zero **What is 10⁰?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1 10⁰ = 1. ::: :::keep-together ### Review: powers **Review: what is 2⁵?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 32 2 × 2 × 2 × 2 × 2 = 32. ::: ## Make: 2⁹ two ways Then write 2⁹ as a quotient of two powers of 2, and as a power of a power. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Simplify 4⁶ ÷ 4² and (3²)⁴, leaving each as a power.** {{lines-3}} ::: :::teacher-note **Answer:** 4⁴ and 3⁸. :::