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# Powers

Number Foundations · Squares, powers and primes

<!-- Speaker note: Students grow factor trees for powers of 2 to see repeated equal factors, read and write exponent notation (base and exponent), evaluate powers, and meet powers of 10 (AC9M7N02, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 2 × 2 × 2 × 2 is written 2⁴: the base 2 is multiplied by itself, and the exponent 4 counts how many times. 2⁴ = 16, not 8.

- I can write a repeated multiplication as a power, and expand a power.
- I can say what the base and the exponent mean.
- I can work out powers, including powers of 10.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need squares (previous lessons) and primes and factor pairs (Year 6). The one idea: the exponent counts how many times the base is multiplied. The common mistake is multiplying base and exponent (2⁵ = 10, Mia's mistake). Fluency with the notation is the hinge for prime factorisation in the next lesson. The factor tree board is used again for prime factors and HCF and LCM. -->

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## Why it matters

> Fold a sheet of paper in half, then again, and again. After 7 folds it is 2⁷ = 128 layers thick. Powers grow fast, and the short way to write them is exponent notation.

<!-- Speaker note: When the same number is multiplied again and again, there is a short way to write it: $$2 \times 2 \times 2 \times 2 = 2^4$$ The **base** (2) is the number being multiplied. The **exponent** (4), also called the index or power, counts how many times it appears. You say "2 to the power of 4". You have met two already: $5^2$ is "5 squared", and $2^3$ is "2 cubed". The board grows a **factor tree**: it splits a number into a factor pair (two numbers that multiply to it), and keeps splitting until every branch is a **prime**, a number with exactly two factors, 1 and itself (2, 3, 5, 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

**What is $2^3$?**

- 5
- 6
- 8
- 23

<!-- Speaker note: What happens: 8. 2³ = 2 × 2 × 2 = 8: three 2s multiplied. 6 is 2 × 3. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 6 multiplied 2 × 3. Ask them to write 2³ the long way. -->

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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%2Fnumber%2Fpowers%2Fstep-2&code=widget%3A+factortree%0Anumber%3A+16%0A&fresh=1}}

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

<!-- Speaker note: Click a number that is not prime (or use the Split buttons), then choose a factor pair. -->

<!-- Speaker note: Teacher: ask how many 2s the tree ended with, and where the 4 in 2⁴ comes from. -->

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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?

* 8. 2³ = 2 × 2 × 2 = 8: three 2s multiplied. 6 is 2 × 3.

<!-- 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>

## Read the power

Here is how to go between the two forms.

Now try one: $3^4$.

$$2 \times 2 \times 2 \times 2 \times 2 = 2^5$$

1) $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$ — Expand and multiply: 4, 8, 16, 32.
2) $3 \times 3 \times 5 = 3^2 \times 5$ — Different bases get their own powers.
3) $10^4 = 10\,000$ — Powers of 10: the exponent counts the zeros.

<!-- 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 × 2 × 2 × 2 × 2 = 2^5 and say "Count the equal factors: five 2s, so the exponent is 5." Line 1: write 2^5 = 2 × 2 × 2 × 2 × 2 = 32 and say "Expand and multiply: 4, 8, 16, 32." Line 2: write 3 × 3 × 5 = 3^2 × 5 and say "Different bases get their own powers." Line 3: write 10^4 = 10000 and say "Powers of 10: the exponent counts the zeros." -->

<!-- Speaker note: Teacher: link 10³ to place value: thousands. -->

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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>

## Read the power

**Work out $3^4$.**

* Answer: 81

<!-- Speaker note: 3 × 3 = 9, × 3 = 27, × 3 = 81. -->

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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

**Mia works out $2^5$. Click the line where her working goes wrong.**

```
 1  Work out 2⁵.
 2  The base is 2 and the exponent is 5.
 3  2⁵ = 2 × 5
 4  = 10
```

* Answer: line 3

<!-- Speaker note: Line 3: 2⁵ is five 2s multiplied, 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5. -->

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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>

## Match them

**Match each power to what it means.**

| | |
|---|---|
| `$10^3$` | A. 3 × 3 = 9 |
| `$3^2$` | B. 10 × 10 × 10 = 1000 |
| `$2^3$` | C. 2 × 2 × 2 = 8 |
| `$5^1$` | D. just 5 |

* Answer: $10^3$ → B. 10 × 10 × 10 = 1000; $3^2$ → A. 3 × 3 = 9; $2^3$ → C. 2 × 2 × 2 = 8; $5^1$ → D. just 5

<!-- Speaker note: The exponent counts the factors. 10³ is 1 followed by three zeros; 5¹ is one 5. -->

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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>

## One change at a time



- **Thirty-two:** Each task changes one thing.
- **Write the power:** 
- **Two bases:** 
- **Swap them:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/powers/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a bigger tree, writing a power from a product, two bases, then the exponent 1 and the order of base and exponent. Hints, in order: Start with 2 × 16, or 4 × 8. For students who need more support: Expand every power before working it out; use the tree board for powers of 2 and 3. For early finishers: The fold-the-paper hook: after how many folds is the stack more than 1000 layers? More than 1 000 000? The hard part, on purpose: Reading the small raised number as a count, not a multiplier. Ask students to write the power the long way first, every time, before working it out. When students get stuck: A student multiplies base and exponent (Mia's mistake). Ask: Write it the long way first. How many 2s? Fix: Expand, then multiply one factor at a time. / A student swaps base and exponent (7³ written 3⁷). Ask: Which number is being multiplied? Fix: The base is the big number; the small raised number counts. / A student adds different bases (2² × 5³ as 7⁵). Ask: Are the factors the same number? Fix: Only equal factors share a power; each base gets its own. -->

<!-- Speaker note: Teacher: for 2⁵ against 5², ask for a prediction before working them out. -->

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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>

## Evaluate

**Work out $4^3$.**

* Answer: 64

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

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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>

## Powers of 10

**Work out $10^5$.**

* Answer: 100000

<!-- Speaker note: 1 followed by five zeros: 100 000. -->

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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>

## Write it

**Write $2 \times 2 \times 5 \times 5 \times 5$ with exponents.**

- $4 \times 15$
- $2^3 \times 5^2$
- $10^5$
- $2^2 \times 5^3$

* Answer: $2^2 \times 5^3$

<!-- Speaker note: Two 2s and three 5s: 2² × 5³ = 4 × 125 = 500. -->

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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: roots

**Review: work out $\sqrt{144}$.**

* Answer: 12

<!-- Speaker note: 12 × 12 = 144. -->

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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: 64 three ways

Then write 64 as a power of 2, a power of 4 and a power of 8.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/powers/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Growing a tree for 64 and writing it as a power of 2, of 4 and of 8 shows one number has several power names, all from the same six 2s. Hints, in order: Split 64 into 8 × 8, or 2 × 32, or 4 × 16. / Group the six 2s in pairs for 4s, in threes for 8s. -->

<!-- Speaker note: Teacher: collect the three names for 64 and ask for a fourth (64¹). -->

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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

**Work out 3³ and write 5 × 5 × 5 × 5 as a power.**

* 3³ = 27; 5⁴.

<!-- 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: Next: Prime factors (node 2), writing any number as a product of prime powers. Where it leads: index laws and scientific notation (Year 8 and 9), and square numbers as powers with exponent 2. -->