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Calculating with roots

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

Multiplying roots means multiplying the radicands: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). And a root can be written as a power: \( \sqrt{a} = a^{\frac{1}{2}} \).
📄 Open worksheet (PDF, German)

The video

Watch the colours: the radicand is purple, 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. 0:15 An example
  3. 0:54 Why is it true?
  4. 1:26 Two irrational numbers
  5. 2:07 Simplifying roots
  6. 2:27 Careful, mistake!
  7. 2:48 The root as a power
  8. 3:28 The n-th root as a power
  9. 4:01 Try it yourself

The most important points

\( \sqrt{4} \cdot \sqrt{9} = 2 \cdot 3 = 6 \) and \( \sqrt{4 \cdot 9} = \sqrt{36} = 6 \). Both ways lead to the same result.

Reason with the 4th law of exponents: (\sqrt{a} · \sqrt{b})^2 = a · b .
Reason with the 4th law of exponents: \( (\sqrt{a} \cdot \sqrt{b})^2 = a \cdot b \).

Multiplying roots

\( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) (for \( a, b \geq 0 \)). Multiplying roots means multiplying the radicands (die Radikanden). Careful: this does not work for plus: \( \sqrt{9 + 16} = 5 \), but \( \sqrt{9} + \sqrt{16} = 7 \).

Two irrational numbers: \sqrt{12} · \sqrt{108} = \sqrt{4 · 36 · 9} = 2 · 6 · 3 = 36 .
Two irrational numbers: \( \sqrt{12} \cdot \sqrt{108} = \sqrt{4 \cdot 36 \cdot 9} = 2 \cdot 6 \cdot 3 = 36 \).

Simplifying roots: \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \), \( \sqrt{108} = 6\sqrt{3} \).

Derivation: (a^x)^2 = a^{2x} = a^1 , so 2x = 1 and x = 1/2 .
Derivation: \( (a^x)^2 = a^{2x} = a^1 \), so \( 2x = 1 \) and \( x = \frac{1}{2} \).

The root as a power

\( \sqrt{a} = a^{\frac{1}{2}} \), \( \sqrt[3]{a} = a^{\frac{1}{3}} \), in general \( \sqrt[n]{a} = a^{\frac{1}{n}} \). Examples: \( 4^{\frac{1}{2}} = 2 \), \( 8^{\frac{1}{3}} = 2 \).

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
The worksheet – Page 5
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Try it yourself

Task

Calculate: a) \( \sqrt{2} \cdot \sqrt{8} \) b) \( \sqrt{3} \cdot \sqrt{27} \) c) \( 9^{\frac{1}{2}} \) d) \( 27^{\frac{1}{3}} \)

Show solution

a) \( 4 \) b) \( 9 \) c) \( 3 \) d) \( 3 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: calculating with roots (root a times root b = root of a times b, the root as a power with a fraction as exponent). 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.