--- title: "Using the index laws" subtitle: "Irrational numbers and index laws · Index laws · Lesson 6" --- # Using the index laws **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Scientists write the distance to the Sun as 1.5 × 10⁸ km. Multiplying numbers like that takes seconds with the index laws, as long as you only join powers of the same base. :::teacher-note **In one sentence:** Students combine the index laws one step at a time, check that the bases match before joining powers, and use the laws with powers of 10 (AC9M8N02, node 2). **Students should be able to say:** I can simplify an expression using more than one index law. I can tell when powers can be joined (same base) and when they cannot. I can multiply numbers in scientific notation using the laws. **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 :::keep-together **Can 2⁵ × 3² be simplified to a single power using the index laws?** ☐ No: the bases are different ☐ Yes: 6¹⁰ ☐ Yes: 6⁷ ☐ Yes: 5⁷ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** No. The laws join powers of the same base only. 2⁵ × 3² = 32 × 9 = 288. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## One step at a time :::keep-together ### Step by step Here is the routine. 1. $(2^3)^2 \times 2^4 \div 2^5$ — Powers of powers first. 2. $= 2^6 \times 2^4 \div 2^5$ — 3 × 2 = 6. 3. $= 2^{10} \div 2^5$ — Then left to right: 6 + 4 = 10. 4. $= 2^5 = 32$ — 10 − 5 = 5. **Simplify 5⁴ × 5³ ÷ 5⁵. 5 to what power?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2 5⁴ × 5³ = 5⁷, then 5⁷ ÷ 5⁵ = 5². ::: :::keep-together ### Find the mistake **Lia simplifies 2⁴ × 5². Click the line where her working goes wrong.** ``` 1 Lia simplifies 2⁴ × 5². 2 Add the indices: 4 + 2 = 6. 3 Multiply the bases: 2 × 5 = 10. 4 So 2⁴ × 5² = 10⁶. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: the indices only add for the same base. 2⁴ × 5² = 16 × 25 = 400, not a million. ::: :::keep-together ### Scientific notation **What is (4 × 10⁵) × (2 × 10²)?** ☐ 8 × 10³ ☐ 6 × 10⁷ ☐ 8 × 10⁷ ☐ 8 × 10¹⁰ ::: :::teacher-note **Answer:** 8 × 10⁷ 4 × 2 = 8 and 10⁵ × 10² = 10⁷. ::: :::typical ## Combinations **A longer one.** **Equal to 64.** **Equal to 1000.** **Which combine?.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Using the laws :::keep-together ### Power then product **Simplify (3²)³ × 3. 3 to what power?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 (3²)³ = 3⁶, and 3⁶ × 3¹ = 3⁷. ::: :::keep-together ### Two divisions **Simplify 6⁸ ÷ 6³ ÷ 6². 6 to what power?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 8 − 3 − 2 = 3: 6³. ::: :::keep-together ### Adding powers **What is 2³ + 2³?** ☐ 2⁶ (64) ☐ 2⁴ (16) ☐ 4³ ☐ 4⁶ ::: :::teacher-note **Answer:** 2⁴ (16) 8 + 8 = 16 = 2⁴. Adding two equal powers doubles one. ::: :::keep-together ### Review: evaluate **Review: what is 4³?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 64 4 × 4 × 4 = 64. ::: ## Make: build an expression Then write an expression using all three laws that simplifies to 2⁵, and swap with a partner to check. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Simplify (5³)² × 5⁴ ÷ 5⁸.** {{lines-3}} ::: :::teacher-note **Answer:** 5⁶ × 5⁴ ÷ 5⁸ = 5¹⁰ ÷ 5⁸ = 5². :::