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The fifth law of exponents

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

Dividing powers with the same exponent: we divide the bases, the exponent stays. \( a^n : b^n = (a : b)^n \). Backwards: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).
📄 Open worksheet (PDF, German)

The video

Watch the colours: the bases are blue and 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. 0:13 An example
  3. 0:46 The fifth law
  4. 1:04 Tricks
  5. 1:30 Powers of fractions
  6. 1:48 Careful, don't mix them up!
  7. 2:07 All five laws
  8. 2:20 Try it yourself

The most important points

\( 6^3 : 2^3 = 216 : 8 = 27 \). As a fraction: on top the six three times, below the two three times. Each pair \( \frac{6}{2} \) is \( 3 \). So \( 3^3 = 27 \).

6^3 : 2^3 = 6/2 · 6/2 · 6/2 = 3^3 = 27
\( 6^3 : 2^3 = \frac{6}{2} \cdot \frac{6}{2} \cdot \frac{6}{2} = 3^3 = 27 \)

5th law of exponents

\( a^n : b^n = (a : b)^n \). Same exponent: we divide the bases, the exponent stays the same. Trick: \( 100^4 : 25^4 = 4^4 = 256 \).

A fraction to the power n : \left(2/3\right)^2 = 2^2/3^2 = 4/9 .
A fraction to the power \( n \): \( \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \).

Careful: in the 2nd law the base is the same, then we subtract the exponents: \( 5^6 : 5^2 = 5^4 \). In the 5th law the exponent is the same, then we divide the bases.

All five laws of exponents at a glance.
All five laws of exponents at a glance.

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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The worksheet – Page 6
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Show the worksheet solutions
The worksheet – Page 7
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Try it yourself

Task

Calculate: a) \( 8^3 : 2^3 \) b) \( 20^4 : 10^4 \) c) \( \left(\frac{2}{5}\right)^2 \) d) \( \left(\frac{1}{2}\right)^3 \)

Show solution

a) \( 4^3 = 64 \) b) \( 2^4 = 16 \) c) \( \frac{4}{25} \) d) \( \frac{1}{8} \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: the fifth law of exponents (a to the n divided by b to the n = (a divided by b) to the n) and powers of fractions. 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.