GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Powers and roots

Quick: what's 2³? If you said 6, you're in good company — and it's wrong. It's 8. Soon you'll see why, and place √50 without a calculator.

Where do square roots live?

Put the roots on the line

√10 means the positive number that gives 10 when you multiply it by itself. Drag each square root to where you think it sits, then check. Stuck? Think about which square numbers are either side.

What a power actually means

2³ — what's going on?

Before the roots make full sense, we need the thing they undo. Here's a power written in index notation: 2³. The big 2 is called the base. The small raised 3 is called the index (you'll also hear it called the power).

Which is closest to what you think 2³ means right now?
How sure are you?

Squares, cubes and their roots

Square, cube, both — or neither?

A square number is a whole number times itself (like 5² = 25). A cube number is a whole number used three times (like 5³ = 125). Sort each number — some belong in both circles.

  • A Square numbers
  • B Cube numbers
  1. 1
  2. 4
  3. 6
  4. 8
  5. 25
  6. 27
  7. 32
  8. 64
  9. 100
  10. 125
  11. 1000

Negative numbers and powers

Positive, negative — or no real answer?

Work out the sign of each answer before you look at the reason.

Still to sort

Positive answer (0)

An even number of negative factors

Where the line is: Negatives multiply in pairs, and each pair makes a positive. With an even count, every negative has a partner.

Negative answer (0)

An odd number of negative factors

Where the line is: With an odd count, one negative is left without a partner, so the answer stays negative.

No real answer (0)

No real number works

Where the line is: Only the square root of a negative lands here. A cube root of a negative is fine, because a negative cubed is negative.

7 of 7 still to sort.

Count the negative signs being multiplied. That's the whole trick.

Order of operations

Find the line that goes wrong

Work out 2 × 3² + √(9 + 16)

A student's answer — which line goes wrong?

Sharpening an estimate

Close in on √40

Without a calculator, find √40 correct to 1 decimal place.

  1. 36 < 40 < 49, so √40 is between 6 and 7.The whole-number bracket — the same move as the number line.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

And once you know your square numbers, you can say roughly where any square root lives — before you touch a calculator.

What you need to know

  • What the base and the index in a power like 2³ each mean
  • Square and cube numbers, and how square roots and cube roots undo them
  • What happens to the sign when you raise a negative number to a power
  • Where powers and roots come in the order of operations
  • How to estimate a square root that isn't a whole number

The big picture

A power is repeated multiplication: in 2³ the base, 2, is multiplied and the index, 3, counts the copies, so 2³ = 2 × 2 × 2 = 8. Square roots and cube roots undo squaring and cubing. Powers and roots come before ×, ÷, + and −, and a root sign over a sum acts like brackets. A negative base gives a positive answer to an even power and a negative one to an odd power. Any square root that isn't whole sits between the roots of the perfect squares either side.

Key points

1In a power like 5³, the base (5) is the number being multiplied and the index (3) counts how many copies: 5³ = 5 × 5 × 5 = 125.
2Square numbers come from n × n (1, 4, 9, 16, 25, …) and cube numbers from n × n × n (1, 8, 27, 64, 125, …). Some numbers, like 1 and 64, are both.
3A root undoes a power: √49 = 7 because 7² = 49, and ∛64 = 4 because 4³ = 64. Check a root by squaring or cubing it.
4The √ sign means the positive square root. A negative number has no real square root, but it does have a cube root: ∛(−8) = −2.
5A negative base gives a positive answer to an even power and a negative answer to an odd power. Brackets matter: (−3)² = 9, but −3² = −9.
6Order of operations: brackets, then powers and roots, then × and ÷, then + and −. A root sign over a sum works like brackets.
7To estimate a square root, find the perfect squares either side: 36 < 45 < 49, so √45 is between 6 and 7. Square decimals to close in further.

Worked example

Problem

Work out 3 × (−2)⁴ − ∛27

⚠ Watch out

Reading the index as a multiplier. 2³ means three 2s multiplied together, 2 × 2 × 2 = 8 — not 2 × 3 = 6. The same slip turns 5² into 10 instead of 25.

🧠

Memory hook

The index is a counter, not a multiplier. The small number up top just counts copies: 2³ is three 2s multiplied together — 8, never 6.

✓

Check yourself

Without a calculator: √70 lies between which two whole numbers, and which one is it nearer? (64 < 70 < 81, so between 8 and 9. 8.5² = 72.25 is over 70, so nearer 8.)

Flashcards

(15)
In 7⁴, which number is the base and which is the index?
7 is the base — the number being multiplied. 4 is the index — how many 7s: 7 × 7 × 7 × 7.
What does 'cubed' mean?
Used three times in a multiplication: 6³ = 6 × 6 × 6 = 216.
The square numbers from 1² to 15²
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
The cube numbers 1³, 2³, 3³, 4³, 5³ and 10³
1, 8, 27, 64, 125 and 1000
What is √144?
12, because 12² = 12 × 12 = 144.
What is ∛1000?
10, because 10³ = 10 × 10 × 10 = 1000.
Is (−6)² positive or negative?
Positive: (−6) × (−6) = 36. Two negatives make a positive.
What is (−2)⁵?
−32. Five negative factors is an odd number, so one is left unpaired and the answer is negative.
What is −7²?
−49. Without brackets the power belongs to the 7 only: square first, then the minus. (−7)² would be 49.
Does √(−25) have a real answer?
No. Squaring any real number never gives a negative, so nothing squares to −25.
What is ∛(−64)?
−4, because (−4) × (−4) × (−4) = −64. Negatives do have cube roots.
Where do powers and roots come in the order of operations?
After brackets, but before × and ÷, and before + and −.
√(64 + 36): root each number, or the sum?
The sum — the root sign acts like brackets. √(64 + 36) = √100 = 10, not 8 + 6 = 14.
Between which two whole numbers is √110?
10 and 11, because 10² = 100 and 11² = 121, and 100 < 110 < 121.
How do you check a square root estimate like √12 ≈ 3.5?
Square it: 3.5² = 12.25, a little over 12 — so √12 is a little less than 3.5.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More Edexcel GCSE Maths topics

How this lesson was checked. This Edexcel GCSE Maths (specification 1MA1)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 30 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.