GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Calculating with roots and indices

√10 isn't a whole number, so is it lost? Not at all. It sits in an exact place between two integers, and you can find it without touching a calculator.

Roots · Number

Where does √10 sit?

A root undoes a square. Use the perfect squares either side of each number (9 and 16, and 121 and 144) to hunt for where its root belongs. Drag √10, √12.25 and √137 to where you think they sit, then check.

Place √10Place √12.25Place √137

Squares and cubes

Square, cube, both or neither?

A square number is an integer multiplied by itself (index 2). A cube number is an integer multiplied by itself three times (index 3). Place each number: square, cube, both or neither. Watch out for the ones you can square but that aren't square numbers themselves.

A Square numbers
B Cube numbers

1
4
5
8
10
25
27
64
100
125
144
1000

Negatives

Negative numbers, powers and roots

Four students are comparing answers about negative numbers with powers and roots.

Which is closest to what you think right now?

√−100 is −10, because 10 × 10 = 100 and you keep the minus sign.A negative number to any power is negative, so (−4)² = −16.You can't take any root of a negative number, so ∛−1000 doesn't exist either.√−100 has no real value, but ∛−1000 = −10, and (−4)² = 16.

How sure are you?

I'm sureI think soI'm guessing

√−100 is −10, because 10 × 10 = 100 and you keep the minus sign.

This is a common misconception

Multiply it back: −10 × −10 = +100, not −100. No real number multiplied by itself gives a negative number, so √−100 has no real value.

A negative number to any power is negative, so (−4)² = −16.

This is a common misconception

(−4)² means −4 × −4, and that is +16. A negative number to an even index is positive. Only an odd index leaves it negative, as in (−4)³ = −64.

You can't take any root of a negative number, so ∛−1000 doesn't exist either.

This is a common misconception

Cube numbers can be negative, because a negative number cubed is negative. (−10)³ = −1000, so ∛−1000 = −10. It's the square root of a negative that has no real value.

√−100 has no real value, but ∛−1000 = −10, and (−4)² = 16.

This matches the evidence

Yes. Squaring can never give a negative, so √−100 has no real value. Cubing keeps the sign of the base, so ∛−1000 = −10. And a negative number to an even index is positive.

No real number multiplied by itself is negative, so √−100 has no real value. A negative number cubed is negative, so ∛−1000 = −10. Even index: positive. Odd index: negative.

Order of operations

Where does this working go wrong?

Work out 5² + 10² ÷ 4

A student's working — which line goes wrong?

5² + 10² ÷ 4
= 25 + 100 ÷ 4
= 125 ÷ 4
= 31.25

Laws of indices

Supply the missing steps

Write (5³ × 5⁴) ÷ (5²)⁴ as a single power of 5.

(5³ × 5⁴) ÷ (5²)⁴Take it one piece at a time. All the bases are 5, so the laws of indices can be used.
5³ × 5⁴ = (5 × 5 × 5) × (5 × 5 × 5 × 5) = 5⁷An index counts how many times the base is multiplied by itself. Three 5s times four 5s makes seven 5s, so multiplying powers of the same base adds the indices: 3 + 4 = 7.
missing step

Which line is step 3?

(5²)⁴ = 5⁸(5²)⁴ = 5⁶(5²)⁴ = 5¹⁶

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A root undoes a power, and even roots that aren't whole numbers have a place you can find.

What you need to know

An index counts how many times the base is multiplied by itself: 2 × 2 × 2 = 2³.
A square number is an integer times itself (index 2). A cube number is three of them multiplied (index 3).
You can square 5, but 5 isn't a square number; 25 is. And 1 and 64 are both square and cube numbers.
Have a goSam insists that 7 is a square number, 'because you can square it'. Which number should Sam have called the square number?
49, because 7 × 7 = 49.
You can square 7, but the name belongs to the result of multiplying an integer by itself. 7 isn't a square number; 49 is.
A root undoes a power: √25 = 5, √144 = 12, ∛8 = 2, ∛125 = 5.
A negative number to an even index is positive, (−4)² = 16. To an odd index it stays negative.
No real number times itself is negative, so √−100 has no real value. But ∛−1000 = −10.
Have a goPredict before you calculate: is (−3)³ positive or negative? Then give its value.
Negative: −27.
The index 3 is odd, so the negative sign stays. −3 × −3 = 9, and 9 × −3 = −27.
Order of operations: powers and roots first, then × and ÷, then + and −. A radical sign over a sum acts as a bracket.
Estimate a root between the perfect squares either side. The midpoint check (3.5² = 12.25) shows √10 is nearer 3 than 4.
Have a go√60 sits between two integers. Which two, and is it nearer the lower or the higher? Work out 7.5² to help.
Between 7 and 8, nearer 8.
49 and 64 are the squares either side of 60. 7.5² = 56.25 and 60 is bigger than that, so √60 is bigger than 7.5.
With the same base: add indices to multiply, subtract to divide, multiply for a power of a power.
Also a⁰ = 1 (a not zero), a⁻ⁿ = 1 ÷ aⁿ, and a^(1/n) is the nth root of a.

The big picture

A root undoes a power. Square and cube numbers come from multiplying a whole number by itself, powers and roots slot into the order of operations, and a root that isn't a whole number still has an exact place between two integers that you can find. The laws of indices handle the rest.

Key points

1Index: how many times the base is multiplied by itself. Square numbers have index 2 and cube numbers index 3. Squares to 15²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Cubes: 1, 8, 27, 64, 125, 1000.

2The name 'square number' or 'cube number' belongs to the result of the repeated multiplication. 1 and 64 are both.

3A root undoes a power. √ of a negative has no real value. ∛ of a negative exists: ∛−1000 = −10.

4A negative number to an even index is positive, and to an odd index negative.

5Powers and roots, then × and ÷, then + and −. A radical bar across a sum is an implicit bracket.

6Estimate a root between the perfect squares either side, then compare with the midpoint square (3.5² = 12.25) to see which integer it is nearer.

7Same-base laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐˣⁿ, a⁰ = 1 (a not zero), a⁻ⁿ = 1 ÷ aⁿ, a^(1/n) = nth root of a.

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Worked example

Problem

Is √12.4 nearer to 3 or to 4?

⚠ Watch out

Writing √−100 = −10. Multiply it back: −10 × −10 = +100, not −100. No real number multiplied by itself is negative, so √−100 has no real value.

🧠

Memory hook

Three and a half squared is twelve and a quarter: 3.5² = 12.25, which is 3 × 4 + 0.25. For a number between 9 and 16, below 12.25 means its root is nearer 3, and above 12.25 means nearer 4.

✓

Check yourself

Without a calculator: √90 sits between which two integers, and which one is it nearer? Use the perfect squares either side, then the midpoint check with 9.5² = 90.25.

Flashcards

(16)

What does the index 3 mean in 2³?

Three 2s multiplied together: 2 × 2 × 2 = 8. The 2 is the base and the 3 is the index.

What do the names 'square number' and 'cube number' describe?

The result of multiplying an integer by itself (square) or by itself three times (cube), not any number you could square or cube.

The square numbers up to 15²

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Common cube numbers

1, 8, 27, 64, 125, 1000

Which numbers are both square and cube numbers?

1 and 64: 64 = 8² and 64 = 4³.

√25, √49 and √144

5, 7 and 12: each is the integer that was multiplied by itself.

∛8 and ∛125

2 and 5: each is the integer that was multiplied by itself three times.

Why does √−100 have no real value?

No real number multiplied by itself gives a negative number.

Why does ∛−1000 exist, and what is it?

A negative number cubed is negative, so cube numbers can be negative. (−10)³ = −1000, so ∛−1000 = −10.

Sign of a negative number to an even index? To an odd index?

Even index: positive, e.g. (−4)² = 16. Odd index: negative.

Order of operations with powers and roots

Powers and roots first, then × and ÷, then + and −.

What does a radical sign stretched across a sum do?

It acts as an implicit bracket: work out everything under it first.

How do you estimate a square root, and decide which integer it is nearer?

Find the perfect squares either side. Then compare with the midpoint square: for √ between 3 and 4, 3.5² = 12.25.

aᵐ × aⁿ and aᵐ ÷ aⁿ (same base)

aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ: add the indices to multiply, subtract to divide.

(aᵐ)ⁿ

aᵐˣⁿ: multiply the indices for a power of a power.

a⁰, a⁻ⁿ and a^(1/n)

a⁰ = 1 (a not zero); a⁻ⁿ = 1 ÷ aⁿ; a^(1/n) is the nth root of a.

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.

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Where does √10 sit?

Square, cube, both or neither?

Negative numbers, powers and roots

Where does this working go wrong?

Supply the missing steps

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