Up with no ceiling
The second power was a shortcut for 7 · 7. But what if the same number is multiplied three times? Or five times?
You write it the same way, only the number on top changes. 2³ = 2 · 2 · 2 = 8. You read it “two cubed” or “two to the power of three”. And 2⁵ = 2 · 2 · 2 · 2 · 2 = 32.
The rule is still one: the exponent says how many times the base stands in the multiplication. No new trick, just a longer chain.
The third power has its own nickname — it is called a cube, because a cube with edge 3 cm has volume 3 · 3 · 3 = 27 cm³. The second power was a square, the third is a cube.
One more thing will surprise you today: powers grow insanely fast. 2¹⁰ is already 1024. Ten twos in a row and you are past a thousand. That is exactly why powers are used where something doubles — like computer memory.
Jonáš forwards a joke to two classmates. Each of them sends it to two more. And each of those sends it to two more. Sofie draws it on paper as a tree: round 1 has 2 people, round 2 has 4, round 3 has 8. “That is times two every time,” she says. “So round 10 is 2¹⁰… 1024 people.” Jonáš stares: “The whole school in ten rounds?” Sofie nods: “That is why people say something spreads like an avalanche. Maths has a shortcut for it — a power.”
Learn the powers of two up to 2¹⁰ by heart: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. You will meet them in computing on every corner.
How to calculate aⁿ
The notation aⁿ: a is the base, n is the exponent. It means n identical factors: a · a · … · a.
Steps:
- Write it out: 3⁴ = 3 · 3 · 3 · 3.
- Multiply from the left in pairs: 3 · 3 = 9, then 9 · 3 = 27, then 27 · 3 = 81.
- Result: 3⁴ = 81.
Do not try to multiply everything at once. One step, a mid-result, the next step.
Two certainties for every base: a¹ = a (one number, no multiplication) and 1ⁿ = 1 (ones do not change when you multiply them).
With a negative base the exponent decides: (−2)² = 4, but (−2)³ = −8. An even exponent gives plus (the minuses cancel in pairs), an odd exponent leaves one extra minus — the result is negative.
The nicest base is ten: 10³ = 1000, 10⁶ = 1,000,000. The exponent = the number of zeros. Remember that; in two lessons we will build the writing of big numbers on it.
Example 1: 2⁶ step by step
2⁶ = 2 · 2 · 2 · 2 · 2 · 2.
In pairs: 2 · 2 = 4 → 4 · 2 = 8 → 8 · 2 = 16 → 16 · 2 = 32 → 32 · 2 = 64.
A faster way: 2⁶ = 2³ · 2³ = 8 · 8 = 64. Splitting a power into two smaller ones is legal — it is still six twos in a multiplication, just grouped differently.
Example 2: (−2)⁴ and (−2)⁵
(−2)⁴ = (−2) · (−2) · (−2) · (−2). Pairs of minuses cancel: the first two give +4, the second two +4. Altogether 4 · 4 = 16. Even exponent → plus.
(−2)⁵ = (−2)⁴ · (−2) = 16 · (−2) = −32. Odd exponent → one minus is left.
A helper: count the minuses. An even number cancels, an odd number does not.
Example 3: the volume of a cube
A cube box has edge 5 cm. What is its volume?
Volume of a cube = edge³ = 5³ = 5 · 5 · 5.
In steps: 5 · 5 = 25, then 25 · 5 = 125 cm³.
Notice the unit: cm³, because three dimensions were multiplied. The second power carries cm², the third cm³. The exponent in the unit tells you how many dimensions were multiplied.
3⁴ is not 3 · 4 = 12. It is 3 · 3 · 3 · 3 = 81. The same trap as with squares, only it hurts more — the gap between 12 and 81 is already a canyon.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.
Calculate the power. Write it out as multiplication on paper; in the box write only the result.
On paper
Feel the powers of two and three with your hands. Then you will recognise them everywhere.
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Write a column of powers of two: 2¹ to 2¹⁰. Calculate each row from the previous one (times two).
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Calculate 3³, 4³ and 10³. For ten, write what the exponent does with the zeros.
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Decide without calculating whether the result is plus or minus: (−3)⁴, (−5)³, (−1)¹⁰⁰. Then check at least the first two by calculating.
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A cube has edge 4 cm. Calculate the volume and write it with the unit.
Now you 💪
- I can write 2⁵ as a multiplication and calculate it in steps.
- I know why (−2)⁴ comes out positive and (−2)⁵ negative.
- With 10ⁿ I can tell the number of zeros from the exponent.
Done when: You have the powers of two up to 2¹⁰ on paper, three cube powers, and a run of five correct in practice.
What to take from this lesson
- aⁿ = n identical factors a in a multiplication. Calculate in steps, not all at once.
- The third power is the volume of a cube: edge³.
- An even exponent with a negative base gives plus, an odd one minus.
- 10ⁿ = a one and n zeros.
- Powers grow like an avalanche: 2¹⁰ = 1024.
© 2026 Ing. Martin Polak / AlgoRhino · コンテンツ利用規約