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What are powers?

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

How often can you fold a sheet of paper? You learn a short way to write long multiplications: the power (die Potenz) with base (die Basis) and exponent (der Exponent).
📄 Open worksheet (PDF, German)

The video

In the video we fold a sheet of paper again and again. Watch the colours: the base is blue, the exponent is orange.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:18 Folding paper
  3. 0:55 Folding seven times
  4. 1:23 Folding a hundred times
  5. 1:47 The new way to write it
  6. 2:21 Two to the power of four
  7. 2:45 Writing as a power
  8. 3:13 Calculating powers
  9. 3:45 Careful, mistake!
  10. 4:11 Try it yourself

The most important points

We fold a sheet of paper in the middle. With every fold the number of layers (die Lage) doubles: \( 2 \), \( 2 \cdot 2 = 4 \), \( 2 \cdot 2 \cdot 2 = 8 \), … After folding 7 times there are 128 layers – as thick as a book.

With every fold we multiply the layers by 2. After 7 folds: 128 layers.
With every fold we multiply the layers by 2. After 7 folds: 128 layers.

And folding 100 times? Then we would have to multiply 100 twos: \( 2 \cdot 2 \cdot 2 \cdot \ldots \cdot 2 \). That is very tedious.

The power (die Potenz)

\( 2^{100} \) is the short way to write 100 twos multiplied together. The number at the bottom is the base (die Basis): it is multiplied. The number at the top is the exponent (der Exponent): it counts how many times the base is there. The operation is called raising to a power (das Potenzieren). In German we say: „zwei hoch hundert“ (two to the power of one hundred).

The base is blue, the exponent is orange.
The base is blue, the exponent is orange.

Examples: \( 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16 \) („zwei hoch vier“), \( 3^2 = 3 \cdot 3 = 9 \), \( 5^3 = 5 \cdot 5 \cdot 5 = 125 \), \( 10^4 = 10 \cdot 10 \cdot 10 \cdot 10 = 10\,000 \). With ten, the exponent counts the zeros.

Careful: 2^3 is not 2 · 3 . You must not swap the base and the exponent.
Careful: \( 2^3 \) is not \( 2 \cdot 3 \). You must not swap the base and the exponent.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
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Try it yourself

Task

Write as a power and calculate: a) \( 3 \cdot 3 \cdot 3 \) b) \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \) c) \( 10 \cdot 10 \cdot 10 \)

Show solution

a) \( 3^3 = 27 \) b) \( 2^6 = 64 \) c) \( 10^3 = 1000 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: powers as a short way of writing (base, exponent, for example 2 to the power of 4 = 2 · 2 · 2 · 2). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.