---
marp: true
theme: default
html: true
---

# Index laws

Irrational numbers and index laws · Index laws

<!-- Speaker note: Students write powers out as factors on the board and discover the product, quotient and power laws and why a⁰ = 1 (AC9M8N02, node 1). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min). -->

---

## Today

**We are learning to** write powers out as factors and the laws appear: multiplying powers of the same base adds the indices, dividing subtracts them, and a power of a power multiplies them. Anything (except 0) to the power 0 is 1.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need exponent notation (Number Foundations: Powers). The one idea: the laws come from counting factors; the base never changes. The common mistake is multiplying the bases (Kai's mistake). The board writes each power as a row of its factors: joined for a product, cancelled in pairs for a quotient, in groups for a power of a power. The quotient board keeps m at least n (negative indices are Year 9). Wider: the pattern 2³, 2², 2¹, 2⁰ (halving each time) also gives 2⁰ = 1, and leads to 2⁻¹ = ½. -->

---

## Why it matters

> 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.

<!-- Speaker note: A power is repeated multiplication: $2^3 = 2 \times 2 \times 2$. The **base** is 2 and the **index** (exponent) is 3. Writing powers out as factors gives the **index laws** for powers of the **same base**: | Law | Why | Example | |---|---|---| | $a^m \times a^n = a^{m+n}$ | the factors join | $2^3 \times 2^2 = 2^5$ | | $a^m \div a^n = a^{m-n}$ | pairs of factors cancel | $3^5 \div 3^2 = 3^3$ | | $(a^m)^n = a^{mn}$ | $n$ groups of $m$ factors | $(2^3)^2 = 2^6$ | | $a^0 = 1$ | $a^n \div a^n = 1$ and $= a^0$ | $7^0 = 1$ | -->

---

<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**2³ × 2⁴ = 2 to what power?**

- 7
- 1
- 12
- It is 4⁷

<!-- Speaker note: What happens: 2⁷: three 2s times four 2s is seven 2s multiplied together. The base stays 2. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask a '12' student to write 2³ × 2⁴ out. -->

---

<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Freal-numbers%2Findex-laws%2Fstep-2&code=widget%3A+indexlaws%0Amode%3A+product%0Aslider+a%3A+2%2C+10%2C+2%0Aslider+m%3A+1%2C+6%2C+2%0Aslider+n%3A+1%2C+6%2C+2%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/index-laws/?full=step-2#step-2)

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask what happens to the count when n goes up by 1. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 2⁷: three 2s times four 2s is seven 2s multiplied together. The base stays 2.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

---

<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Divide

Here is the routine.

$$3^5 \div 3^2 = \frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$$

1) $= 3 \times 3 \times 3 = 3^3$ — Two pairs cancel: 5 − 2 = 3 factors left.
2) $(2^3)^2 = 2^3 \times 2^3 = 2^6$ — A power of a power: 2 groups of 3 factors, 3 × 2.
3) $2^3 \div 2^3 = 1 = 2^0$ — Everything cancels: so 2⁰ = 1.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 3^5 ÷ 3^2 = 3 × 3 × 3 × 3 × 3 ÷ 3 × 3 and say "Write both powers out." Line 1: write = 3 × 3 × 3 = 3^3 and say "Two pairs cancel: 5 − 2 = 3 factors left." Line 2: write (2^3)^2 = 2^3 × 2^3 = 2^6 and say "A power of a power: 2 groups of 3 factors, 3 × 2." Line 3: write 2^3 ÷ 2^3 = 1 = 2^0 and say "Everything cancels: so 2⁰ = 1." -->

<!-- Speaker note: Teacher: ask Kai how many 3s there are. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 27px; }</style>

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Divide

**5⁶ ÷ 5² = 5 to what power?**

* Answer: 4

<!-- Speaker note: Two pairs of 5s cancel, leaving four: 5⁴. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.
```

* Answer: line 3

<!-- Speaker note: Line 3: the base stays 3. There are six 3s multiplied together: 3⁶ (= 729), not 9⁶. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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

* Answer: 5³ ÷ 5³ = 1, and the quotient law gives 5³⁻³ = 5⁰

<!-- Speaker note: Dividing anything by itself gives 1, and the law says it is a⁰. -->

---

<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Each law on the board



- **Quotient:** 
- **Power of a power:** 
- **Zero index:** 
- **Power or product:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/index-laws/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Building a quotient, a power of a power and a zero index on the board, then telling the product and power laws apart. For students who need more support: A card for each law with the expanded-form picture beside it. For early finishers: Continue the pattern 2⁴, 2³, 2², 2¹, 2⁰ by halving. What should 2⁻¹ and 2⁻² be? The hard part, on purpose: Keeping the base. Ask students to write the powers out as factors before they use any law. When students get stuck: A student multiplies the bases. Ask: Write both powers out: what number is repeated? Fix: The base stays the same; count the factors. / A student multiplies indices for a product. Ask: How many factors altogether? Fix: Products add; powers of powers multiply. / A student says a⁰ = 0. Ask: What is 5³ ÷ 5³? Fix: 1, and the law calls it 5⁰. -->

<!-- Speaker note: Teacher: ask how (2²)³ differs from 2² × 2³. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Product

**7⁵ × 7³ = 7 to what power?**

* Answer: 8

<!-- Speaker note: 5 + 3 = 8: 7⁸. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Power of a power

**(4²)⁵ = 4 to what power?**

* Answer: 10

<!-- Speaker note: 5 groups of 2: 2 × 5 = 10, so 4¹⁰. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Zero

**What is 10⁰?**

* Answer: 1

<!-- Speaker note: 10⁰ = 1. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: powers

**Review: what is 2⁵?**

* Answer: 32

<!-- Speaker note: 2 × 2 × 2 × 2 × 2 = 32. -->

---

<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: 2⁹ two ways

Then write 2⁹ as a quotient of two powers of 2, and as a power of a power.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/index-laws/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Finding a product that makes a target power uses the product law backwards. Hints, in order: The indices must add to 9. / With m bigger than n: 5 and 4, or 6 and 3. -->

<!-- Speaker note: Teacher: ask for all the ways they found. -->

---

<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Simplify 4⁶ ÷ 4² and (3²)⁴, leaving each as a power.**

* 4⁴ and 3⁸.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This starts AC9M8N02. Where it leads: combining the laws (next lesson), scientific notation, and negative indices (Year 9). -->