Maths · KS3 and GCSE · Interactive lesson
Powers, square roots and standard form
Count repeated factors, connect square roots with square areas and use powers of ten to write large and small numbers.
Choose your view
Explore an interactive example, then try the questions at your own pace.
KS3 / GCSE · start with the idea
By the end of this lesson: Explain a power as repeated multiplication.
The interactive diagram and controls are in Stage 2, “Explore one example”. Read the idea here, then try it next.
- A power tells you how many equal factors to multiply.
- 3³ means 3 × 3 × 3, not 3 × 3.
- The index is the small raised number. Increase it by 1 to multiply by the base once more.
Useful starting points
These lessons open in a new tab so you can return here.
What do the words mean?
- Base
- The repeated factor, such as 3 in 3⁴.
- Index (power)
- How many times the base is a factor for a positive whole-number index.
- Square root
- The non-negative number whose square is the given number.
- Standard form
- A number written as a × 10ⁿ with 1 ≤ a < 10.
Change the example
This is the lesson example. Your practice question uses its own numbers.
Something to try
Keep the base the same. Increase the index by 1. What number multiplies the previous answer?
Notice what changes
How particle motion works +
Starts paused. Press Play, then add energy to see the warmer particles move faster.
Particles keep the same size. Liquid particles move around; solid particles vibrate around fixed positions. This is a simplified model, with the change in motion exaggerated to make it easier to see.
About the playback +
About this model
The examples use positive base numbers. Standard form is another way of writing the same value, not a change in the quantity.
More about this diagram
Still picture. Play particle motion in Stage 2.
3³ = 27
Still picture of the example from Stage 2. Use the back arrow to change it.
Work through it
Starting example: read the steps below
- The base is 3; the index is 3.
- Write 3 factors of 3.
- Multiply 3 × 3 × 3.
- The value is 27.
Try on your own
Your turn. No rush.
Question 1
Calculate 3³.
About these questions
Questions appear in a mixed order. Work at your own pace and stop whenever you like.
These questions use their own numbers, separate from the lesson example. You can stop whenever you like. Answers last for this visit to the page.
A good place to pause
This recap is for the activity you just tried. You can return to other activities in this topic later.
Can you explain what the index or square-root symbol tells you to do?
Try one question and explain your first step. If you need help, use a hint, go back to the idea, or ask Maurice. You choose the pace.
Explain it in your own words and add a small example. This can help you remember and spot what to ask Maurice.
How my note is saved
Your note stays in this browser. Other people using this browser can read it. Clearing browser data removes it.
Up to 1,000 characters. A sentence or two is enough.
Your note saves automatically as you type.
Save or revise this activity
Choose to save this activity and a suggested revisit date in this browser. Your answers are not saved.
Try three questions on this activity →
Useful starting points
Share this lesson using its page address. Answers reset when you reopen it; your topic labels stay in this browser.
Notes for parents and tutors
These lessons cover selected KS3 and GCSE ideas for England; they are not a complete course. The level notes explain where GCSE methods or optional extensions begin. Maurice can help you choose material for your course and exam board.
The content draws on the DfE Maths curriculum, DfE Science curriculum and AQA GCSE Physics specification. Level labels are guidance, not exam-board endorsements. For AQA Radioactivity, calculating net decline as a ratio is Higher-tier work. Net means the overall change; ask Maurice whether this extension is relevant to your course. Explanations and practice were reviewed during development on 21 October 2026; Maurice should check their fit before assigning work.
Topic labels and your lesson-view preference are saved in this browser. This tab also remembers the last lesson stage, so you can choose to continue there. If you choose revision, selected activities and suggested revisit dates are saved here too. Recent question codes are also kept to reduce repeats. Reflection notes are saved automatically per topic when you type them; you can delete a note in the recap. They stay in this browser and are visible to other people using it. Practice answers and enquiry details are not saved. Clearing browser data removes the labels.