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

Powers, roots and exact answers

Index notation is shorthand for repeated multiplication, and roots undo it. The Year 8 step is knowing when an answer is exact and when your calculator has quietly rounded it for you.

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

  • • What an index really means
  • • Roots undo powers
  • • Exact answers versus decimals
  • • Truncating and rounding

This lesson at a glance

  • 30 minutes
  • 19 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 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Powers, roots and exact answers

An English teaching video on this topic from 1st Class Maths. 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

What you already know

Part 2 of 19

Before we start

What you already know

You can multiply and divide integers and know your square numbers to 12².

3

Key words

Part 3 of 19

Key words

Words you'll need today

index
The small raised number showing how many times to multiply.
root
The number that was multiplied by itself to give a result.
exact
Written in a form with nothing rounded away, such as √2 or 1/3.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Part 1 — Powers and roots

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

5

Explore the idea

Part 5 of 19

Learn

What an index really means

2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the index is 5. Powers of 2 (2, 4, 8, 16, 32, 64, 128), of 3 (3, 9, 27, 81), of 4 (4, 16, 64) and of 5 (5, 25, 125, 625) are worth knowing on sight, because they turn up in ratio, probability and computing work.

6

Explore the idea

Part 6 of 19

Learn

Roots undo powers

√49 = 7 because 7² = 49. The cube root ∛64 = 4 because 4³ = 64. Every positive number has two square roots, +7 and −7, though √ on its own means the positive one. Roots and powers are inverse operations, so you can always check a root by squaring or cubing your answer.

7

Quick check

Part 7 of 19

Quick check

What is 3⁴?

8

Quick check

Part 8 of 19

Quick check

∛125 =

9

Reset break

Part 9 of 19

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
10

Learning cycle

Part 10 of 19

Learning cycle 2 of 2

Part 2 — Exact or rounded?

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

11

Explore the idea

Part 11 of 19

Learn

Exact answers versus decimals

√2 = 1.41421356… and never stops or repeats, so 1.41 is not √2 — it is a rounded version. Leaving the answer as √2 is exact. Similarly 1/3 is exact but 0.333 is not. In exam-style questions, 'give your answer in exact form' means keep the surd or the fraction.

12

Explore the idea

Part 12 of 19

Learn

Truncating and rounding

Truncating chops digits off: 1.41421 truncated to 2 d.p. is 1.41. Rounding looks at the next digit first: 1.4159 rounds to 1.42 but truncates to 1.41. Calculators display truncated or rounded values, so two people can get slightly different final answers if they round part-way through. Round only at the end.

13

Quick check

Part 13 of 19

Quick check

Which is an exact answer?

14

Quick check

Part 14 of 19

Quick check

2.7182 truncated to 2 d.p. is...

15

Common mix-ups

Part 15 of 19

Common mix-ups

Things people often get wrong

  • Reading 2⁵ as 2 × 5 = 10 instead of 32.
  • Thinking 1.41 is exactly √2.
  • Rounding in the middle of a calculation and carrying the error forward.
16

Challenge round

Part 16 of 19

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

17

Challenge round

Part 17 of 19

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

____ of 2 (2, 4, 8, 16, 32, 64, 128), of 3 (3, 9, 27, 81), of 4 (4, 16, 64) and of 5 (5, 25, 125, 625) are worth knowing on sight, because they turn up in ratio, probability and computing work.

18

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Powers, roots and exact answers

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

  1. 1. 2⁶ =

  2. 2. √81 =

  3. 3. Which is a power of 5?

  4. 4. π written as 3.14 is...

  5. 5. Why round only at the end?

  6. 6. 5² × 5 =

19

Lesson round-up

Part 19 of 19

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

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 19 · Watch & discover

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