Exponents: Notation, Squares, Cubes & Roots

What exponential notation actually means, how to read and write it, squares, cubes, square roots and cube roots, and scientific notation — with a labelled diagram, worked examples and a quiz.

Exponents are just a shorthand for repeated multiplication. Instead of writing 2 × 2 × 2 × 2 × 2, you write 2⁵. That small raised number saves an enormous amount of writing — and once you meet very large numbers in science, it becomes essential.

1Reading exponential notation

In 2⁵ the big number is the base and the small raised number is the exponent (also called the index).

aⁿ = a × a × a × … × a (n factors)

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The most common error: thinking 2⁵ means 2 × 5 = 10. It does not — it means 2 multiplied by itself 5 times, which is 32.

2Squares and cubes

A number to the power 2 is squared; to the power 3 is cubed. These come from area and volume: a square of side 4 has area 4² = 16, and a cube of side 4 has volume 4³ = 64.

Worked ExampleCalculate 3⁴ and 5³

1
3⁴ = 3 × 3 × 3 × 3.
Write it out as repeated multiplication first.
2
= 9 × 3 × 3 = 27 × 3 = 81.
Work left to right in steps — safer than trying it in one go.
3
5³ = 5 × 5 × 5 = 125.
Cubed means three factors.

3Square roots and cube roots

A root undoes a power. 36 asks "what number squared gives 36?" and ³27 asks "what number cubed gives 27?"

Worked ExampleCalculate 81 and ³64

1
81: which number times itself gives 81?
Think of your square numbers: 8²=64, 9²=81.
2
81 = 9.
Because 9 × 9 = 81.
3
³64: which number cubed gives 64? 4 × 4 × 4 = 64, so ³64 = 4.
Cube roots undo cubing.

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Learn these by heart — they appear in every exam: squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and cubes 1, 8, 27, 64, 125.

4Scientific notation

Scientific notation writes a big number as a number between 1 and 10 multiplied by a power of 10. In Grade 8 the exponents are positive.

Worked ExampleWrite 47 000 in scientific notation

1
Place the decimal comma after the first digit: 4,7
The first part must be between 1 and 10.
2
Count how many places the comma moved: 4 places.
From 47 000, → 4,7 is 4 jumps left.
3
47 000 = 4,7 × 10⁴.
The number of jumps becomes the exponent of 10.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Write 7 × 7 × 7 × 7 in exponential form.
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7⁴
2
Calculate 2⁶.
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2×2×2×2×2×2 = 64
3
Calculate 10³.
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10 × 10 × 10 = 1 000
4
Calculate 144.
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12 (12 × 12 = 144)
5
Calculate ³125.
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5 (5 × 5 × 5 = 125)
6
Calculate 3² + 4².
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9 + 16 = 25
7
Write 3 200 000 in scientific notation.
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3,2 × 10⁶
8
Write 5,4 × 10³ as an ordinary number.
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5 400