Good Learning Co.
SavannahYear 8
← Maths
MathsNumber, powers and accuracy30 min★★★ difficulty

Powers, roots and exact values

A power is repeated multiplication written efficiently, and a root undoes it. Knowing the small powers by heart turns slow arithmetic into instant recognition.

Part of your national curriculum
  • Number: Use integer powers and associated real roots, recognise powers of 2, 3, 4 and 5, and distinguish between exact representations and truncated or rounded decimals

Lesson overview

What you'll learn in this lesson

Use integer powers and associated real roots, recognise powers of 2, 3, 4 and 5, and distinguish between exact representations and truncated or rounded decimals

Key learning points

  • • Powers you should recognise instantly
  • • Roots undo powers
  • • Index laws in practice
  • • Exact versus rounded

This lesson at a glance

  • 30 minutes
  • 17 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

integerpowersassociateddistinguishrepresentations

Scroll down — the lesson carries on below

Part 1 of 17 · Discover6%
1

Watch & discover

Part 1 of 17

Watch first

Powers, roots and exact values

An English teaching video on this topic from Smile and Learn - English. Watch it, then scroll down for the lesson.

In a nutshell

Use integer powers and associated real roots, recognise powers of 2, 3, 4 and 5, and distinguish between exact representations and truncated or rounded decimals

2

Learning cycle

Part 2 of 17

Learning cycle 1 of 2

Part 1 · Powers you should recognise instantly

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 17

Learn

Powers you should recognise instantly

Powers of 2 run 2, 4, 8, 16, 32, 64, 128, 256. Powers of 3 run 3, 9, 27, 81, 243. Powers of 4 run 4, 16, 64, 256, and powers of 5 run 5, 25, 125, 625. Recognising 64 as both 2⁶ and 4³ and 8² lets you rewrite a calculation in whichever base makes it simplest.

4

Explore the idea

Part 4 of 17

Learn

Roots undo powers

The square root of 49 is 7 because 7² = 49; the cube root of 125 is 5 because 5³ = 125. Every positive number has two square roots, one positive and one negative, but the square root symbol means the positive one unless a question says otherwise. Cube roots of negative numbers exist: the cube root of −8 is −2.

5

Quick check

Part 5 of 17

Quick check

Which is 2⁶?

6

Quick check

Part 6 of 17

Quick check

Simplify 5⁷ ÷ 5⁴.

7

Learning cycle

Part 7 of 17

Learning cycle 2 of 2

Part 2 · Index laws in practice

A short piece of teaching, then a check to make sure it has landed.

8

Reset break

Part 8 of 17

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
9

Explore the idea

Part 9 of 17

Learn

Index laws in practice

Multiplying powers of the same base adds the indices, so 2³ × 2⁴ = 2⁷. Dividing subtracts them, so 5⁶ ÷ 5² = 5⁴. A power of a power multiplies them, so (3²)⁴ = 3⁸. Any non-zero number to the power zero equals 1, because dividing a power by itself leaves 1.

10

Explore the idea

Part 10 of 17

Learn

Exact versus rounded

√2 = 1.414213… never terminates, so writing 1.41 changes the value. An exact representation keeps the root, the fraction or π; a rounded value is an approximation you should introduce only at the final step. Truncating cuts digits off, which always under-states a positive number, while rounding chooses the nearer value.

11

Quick check

Part 11 of 17

Quick check

The cube root of 125 is

12

Quick check

Part 12 of 17

Quick check

Which is an exact value for the diagonal of a 1 cm square?

13

Explore the idea

Part 13 of 17

Worked example

Worked answer: simplify 2⁵ × 2³ ÷ 2⁶ and explain each step

Multiplying gives 2⁵⁺³ = 2⁸ because the bases match and repeated multiplication combines (1). Dividing subtracts the indices: 2⁸⁻⁶ = 2² (1). Evaluating gives 4 (1). Writing 2⁵ × 2³ = 4⁸ would be wrong, because the index laws combine indices, never the bases.

14

Challenge round

Part 14 of 17

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

15

Challenge round

Part 15 of 17

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

____ of 2 run 2, 4, 8, 16, 32, 64, 128, 256.

16

Mastery quiz

Part 16 of 17

Marked quiz

End of lesson quiz: Powers, roots and exact values

4 questions, marked with the reasoning shown. No timer.

  1. 1. Which is 2⁶?

  2. 2. Simplify 5⁷ ÷ 5⁴.

  3. 3. The cube root of 125 is

  4. 4. Which is an exact value for the diagonal of a 1 cm square?

17

Lesson round-up

Part 17 of 17

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

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Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 17 · Watch & discover

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    Use integer powers and associated real roots, recognise powers of 2, 3, 4 and 5, and distinguish between exact representations and truncated or rounded decimals

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