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Laws of exponents

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

With the laws of exponents (die Potenzgesetze) you write calculations in a shorter way: multiply and divide powers with the same base, raise a power in brackets to a power, multiply powers with the same exponent.
📄 Open worksheet (PDF, German)

The video

Watch the colours: the base is blue, the exponent is orange. A second base is purple.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:23 Same base: multiply
  3. 0:58 The first law
  4. 1:37 A trick
  5. 2:13 Same base: divide
  6. 2:42 The second law
  7. 3:25 Power of a power
  8. 4:02 Same exponent
  9. 4:29 The fourth law
  10. 5:09 All laws
  11. 5:36 Try it yourself

The most important points

With \( 2 + 5 + 8 \) we change the order: \( 2 + 8 + 5 = 10 + 5 = 15 \). That makes it easier. There are tricks like this for powers too: the laws of exponents (die Potenzgesetze).

\( 3^2 \cdot 3^3 = (3 \cdot 3) \cdot (3 \cdot 3 \cdot 3) = 3^5 \). The three is there \( 2 + 3 = 5 \) times.

1st law (1. Potenzgesetz)

Same base, multiply: the base stays the same. We add the exponents: \( a^m \cdot a^n = a^{m+n} \). Trick: \( 8 \cdot 4 \cdot 16 = 2^3 \cdot 2^2 \cdot 2^4 = 2^9 = 512 \).

Same base: we add the exponents.
Same base: we add the exponents.

\( 4^5 : 4^3 \): we write a fraction and cancel a four three times. At the top \( 5 - 3 = 2 \) fours are left: \( 4 \cdot 4 = 4^2 \).

We cancel one four at the top and one at the bottom, three times.
We cancel one four at the top and one at the bottom, three times.

2nd law (2. Potenzgesetz)

Same base, divide: the base stays the same. We subtract the exponents: \( a^m : a^n = a^{m-n} \). Trick: \( 81 : 27 = 3^4 : 3^3 = 3^1 = 3 \).

3rd law (3. Potenzgesetz)

A power is in brackets: we multiply the exponents: \( (a^m)^n = a^{m \cdot n} \). Example: \( (2^3)^2 = 2^3 \cdot 2^3 = 2^6 \).

4th law (4. Potenzgesetz)

Same exponent: the exponent stays the same. We multiply the bases: \( a^n \cdot b^n = (a \cdot b)^n \). Example: \( 2^4 \cdot 3^4 = 6^4 \). Trick: \( 2^3 \cdot 5^4 = (2 \cdot 5)^3 \cdot 5 = 1000 \cdot 5 = 5000 \).

All four laws. Careful: 3^2 · 3^3 is not 9^5 , but 3^5 .
All four laws. Careful: \( 3^2 \cdot 3^3 \) is not \( 9^5 \), but \( 3^5 \).

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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Show the worksheet solutions
The worksheet – Page 8
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Try it yourself

Task

Write as one power: a) \( 2^5 \cdot 2^3 \) b) \( 7^6 : 7^4 \) c) \( (5^2)^3 \) d) \( 2^3 \cdot 5^3 \)

Show solution

a) \( 2^8 \) b) \( 7^2 \) c) \( 5^6 \) d) \( 10^3 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: laws of exponents (same base: add or subtract the exponents; power in brackets: multiply the exponents; same exponent: multiply the bases). 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.