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Roots

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

We undo raising to a power: taking the root (das Wurzelziehen). You learn how to write it with the root sign, the radicand and the index – from the square root to the n-th root.
📄 Open worksheet (PDF, German)

The video

Watch the colours: the radicand is purple, the index is orange, the result is green.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:14 A square
  3. 0:38 How to write it
  4. 1:14 Examples
  5. 1:36 A cube
  6. 2:07 The index
  7. 2:39 More examples
  8. 3:00 Careful, mistake!
  9. 3:21 Not always whole
  10. 3:34 Try it yourself

The most important points

Minus undoes plus, division undoes multiplication. Taking the root undoes raising to a power. A square has 25 boxes. How long is one side? \( 5 \cdot 5 = 25 \), so the side is 5: \( \sqrt{25} = 5 \).

A square with 25 boxes has side length 5.
A square with 25 boxes has side length 5.

The square root (die Quadratwurzel)

\( \sqrt{\phantom{a}} \) is the root sign (das Wurzelzeichen), the number under it is the radicand (der Radikand). \( \sqrt{a} = b \), because \( b \cdot b = a \) (\( b \) without minus). \( 5^2 \) is also called “five squared” (fünf zum Quadrat), so \( \sqrt{25} \) is the square root. Examples: \( \sqrt{9} = 3 \), \( \sqrt{49} = 7 \), \( \sqrt{100} = 10 \).

A cube made of 27 small cubes has edge length 3: \sqrt[3]{27} = 3 .
A cube made of 27 small cubes has edge length 3: \( \sqrt[3]{27} = 3 \).

The n-th root

The small number at the top left is the index (der Wurzelexponent). \( \sqrt[n]{a} = b \), because \( b^n = a \). For the square root the index is 2; we usually don't write it. Examples: \( \sqrt[3]{8} = 2 \), \( \sqrt[3]{1000} = 10 \), \( \sqrt[4]{16} = 2 \).

Root sign, radicand and index.
Root sign, radicand and index.

Careful: the root is not “divided by 2”. \( \sqrt{16} = 4 \), not 8. Not every root is a whole number: \( \sqrt{2} \approx 1.41 \). You calculate such roots with the calculator.

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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The worksheet – Page 5
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Show the worksheet solutions
The worksheet – Page 6
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Try it yourself

Task

Take the root: a) \( \sqrt{36} \) b) \( \sqrt{81} \) c) \( \sqrt[3]{64} \) d) \( \sqrt[3]{125} \)

Show solution

a) \( 6 \) b) \( 9 \) c) \( 4 \) d) \( 5 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: roots (square root, cube root, radicand, index; the root undoes raising to a power). 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.