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
- 21 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 21
Visual introductionPicture this
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.
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
Learning cycle
Part 2 of 21
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.
Explore the idea
Part 3 of 21
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.
Reset break
Part 4 of 21
Pause
That's sitting 1 of 5 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.
Explore the idea
Part 5 of 21
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.
Quick check
Part 6 of 21
Quick check
Part 7 of 21
Reset break
Part 8 of 21
Pause
That's sitting 2 of 5 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.
Learning cycle
Part 9 of 21
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.
Explore the idea
Part 10 of 21
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.
Explore the idea
Part 11 of 21
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.
Reset break
Part 12 of 21
Pause
That's sitting 3 of 5 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.
Quick check
Part 13 of 21
Quick check
Part 14 of 21
Explore the idea
Part 15 of 21
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.
Reset break
Part 16 of 21
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 21
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
Challenge round
Part 18 of 21
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.
Challenge round
Part 19 of 21
Game · Recall cards
Evaluate 18 − 2 × (3² − 5).
Card 1 of 4
Mastery quiz
Part 20 of 21
Marked quiz
End of lesson quiz: Powers, roots and exact values
4 questions, marked with the reasoning shown. No timer.
1. Evaluate 18 − 2 × (3² − 5).
2. What is the reciprocal of 2/3?
3. Evaluate 5 + 3 × 4.
4. What is 2⁵?
Lesson round-up
Part 21 of 21
Lesson round-up
Ready when you are
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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