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# Using the index laws

Irrational numbers and index laws · Index laws

<!-- Speaker note: 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). About 35 minutes (Predict and Run 8 min · Investigate 10 min · Modify 8 min · Practise and Make 9 min). -->

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## Today

**We are learning to** simplify one step at a time: powers of powers first, then multiply and divide from left to right, keeping each base. The laws only join powers of the same base: 2³ × 3² cannot become one power. They also make scientific notation quick.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the index laws (previous lesson). The one idea: apply one law at a time (powers of powers, then multiply and divide left to right), and only for powers of the same base. The common mistake is joining different bases (Lia's mistake). Adding powers (2³ + 2³) is not covered by the laws: a good discussion. Wider: dividing in scientific notation; a systematic order for long expressions. -->

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## Why it matters

> 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.

<!-- Speaker note: Simplify an expression with several powers one law at a time, keeping track of each base: $$(2^3)^2 \times 2^4 \div 2^5 = 2^6 \times 2^4 \div 2^5 = 2^{10} \div 2^5 = 2^5$$ The laws only work for powers of the **same base**. $2^3 \times 3^2 = 8 \times 9 = 72$: it is not $6^5$, and it cannot be written as one power of a whole number by the laws. In **scientific notation**, the powers of 10 follow the laws: $(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7$. -->

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<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

**Can 2⁵ × 3² be simplified to a single power using the index laws?**

- No: the bases are different
- Yes: 6¹⁰
- Yes: 6⁷
- Yes: 5⁷

<!-- Speaker note: What happens: No. The laws join powers of the same base only. 2⁵ × 3² = 32 × 9 = 288. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask a '6⁷' student to work out 2⁵ × 3² as numbers. -->

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<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%2Fusing-index-laws%2Fstep-2&code=widget%3A+indexlaws%0Amode%3A+product%0Aa%3A+3%0Aslider+m%3A+1%2C+6%2C+1%0Aslider+n%3A+1%2C+6%2C+1%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask what all the ways have in common. -->

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<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?

* No. The laws join powers of the same base only. 2⁵ × 3² = 32 × 9 = 288.

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

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<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>

## Step by step

Here is the routine.

$$(2^3)^2 \times 2^4 \div 2^5$$

1) $= 2^6 \times 2^4 \div 2^5$ — 3 × 2 = 6.
2) $= 2^{10} \div 2^5$ — Then left to right: 6 + 4 = 10.
3) $= 2^5 = 32$ — 10 − 5 = 5.

<!-- 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 (2^3)^2 × 2^4 ÷ 2^5 and say "Powers of powers first." Line 1: write = 2^6 × 2^4 ÷ 2^5 and say "3 × 2 = 6." Line 2: write = 2^10 ÷ 2^5 and say "Then left to right: 6 + 4 = 10." Line 3: write = 2^5 = 32 and say "10 − 5 = 5." -->

<!-- Speaker note: Teacher: ask Lia to check with numbers. -->

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<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>

## Step by step

**Simplify 5⁴ × 5³ ÷ 5⁵. 5 to what power?**

* Answer: 2

<!-- Speaker note: 5⁴ × 5³ = 5⁷, then 5⁷ ÷ 5⁵ = 5². -->

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<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

**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.
```

* Answer: line 2

<!-- Speaker note: Line 2: the indices only add for the same base. 2⁴ × 5² = 16 × 25 = 400, not a million. -->

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<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>

## Scientific notation

**What is (4 × 10⁵) × (2 × 10²)?**

- 8 × 10³
- 6 × 10⁷
- 8 × 10⁷
- 8 × 10¹⁰

* Answer: 8 × 10⁷

<!-- Speaker note: 4 × 2 = 8 and 10⁵ × 10² = 10⁷. -->

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<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>

## Combinations



- **A longer one:** 
- **Equal to 64:** 
- **Equal to 1000:** 
- **Which combine?:** 

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

<!-- Speaker note: What this step is for: Longer combinations, a power of a power to a given value, and choosing which expressions the laws can join. Hints, in order: 64 = 2⁶. / Which m and n multiply to 6? For students who need more support: Write each expression out as factors for the first two problems. For early finishers: Multiply (4 × 10⁵) × (5 × 10³) and give the answer in scientific notation (careful: 4 × 5 = 20). The hard part, on purpose: Checking the bases first. Ask students to circle each base before they use any law. When students get stuck: A student adds indices for different bases. Ask: Circle the bases: are they the same? Fix: Only the same base combines. / A student adds when there is a power of a power. Ask: How many groups, of how many factors? Fix: Powers of powers multiply the indices. / A student treats 3 as 3⁰. Ask: How many 3s is 3? Fix: 3 = 3¹. -->

<!-- Speaker note: Teacher: ask why 7³ + 7⁴ is not 7⁷. -->

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<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 then product

**Simplify (3²)³ × 3. 3 to what power?**

* Answer: 7

<!-- Speaker note: (3²)³ = 3⁶, and 3⁶ × 3¹ = 3⁷. -->

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<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>

## Two divisions

**Simplify 6⁸ ÷ 6³ ÷ 6². 6 to what power?**

* Answer: 3

<!-- Speaker note: 8 − 3 − 2 = 3: 6³. -->

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<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>

## Adding powers

**What is 2³ + 2³?**

- 2⁶ (64)
- 2⁴ (16)
- 4³
- 4⁶

* Answer: 2⁴ (16)

<!-- Speaker note: 8 + 8 = 16 = 2⁴. Adding two equal powers doubles one. -->

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<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: evaluate

**Review: what is 4³?**

* Answer: 64

<!-- Speaker note: 4 × 4 × 4 = 64. -->

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<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: build an expression

Then write an expression using all three laws that simplifies to 2⁵, and swap with a partner to check.

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

<!-- Speaker note: What this step is for: Writing an expression that simplifies to a target uses the laws backwards and in combination. Hints, in order: 256 = 2⁸. / Which m and n multiply to 8? -->

<!-- Speaker note: Teacher: ask a partner to simplify their expression. -->

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<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 (5³)² × 5⁴ ÷ 5⁸.**

* 5⁶ × 5⁴ ÷ 5⁸ = 5¹⁰ ÷ 5⁸ = 5².

<!-- 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 completes AC9M8N02. Where it leads: scientific notation in science, negative indices and algebraic powers (Year 9). -->