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Negative base and order of operations

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

With a negative base the exponent decides the sign: even → positive, odd → negative. And: \( (-2)^4 = 16 \), but \( -2^4 = -16 \). Finally: brackets, powers, multiply/divide, add/subtract.
📄 Open worksheet (PDF, German)

The video

Watch the colours: positive results are green, negative ones are pink.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:16 Negative base
  3. 0:45 Even and odd
  4. 1:07 Careful, brackets!
  5. 1:31 Order of operations
  6. 2:04 Another example
  7. 2:21 Try it yourself

The most important points

\( (-2)^2 = 4 \), \( (-2)^3 = -8 \), \( (-2)^4 = 16 \). The sign changes at every step.

Minus times minus gives plus: the minus signs form pairs.
Minus times minus gives plus: the minus signs form pairs.

Negative base

Even exponent: positive result. Odd exponent: negative result. \( (-1)^{100} = 1 \), \( (-1)^{101} = -1 \).

The exponent belongs only to the number directly in front of it: -2^4 = -16 , but (-2)^4 = 16 .
The exponent belongs only to the number directly in front of it: \( -2^4 = -16 \), but \( (-2)^4 = 16 \).

Order of operations

1. brackets, 2. powers, 3. multiply and divide (die Punktrechnung), 4. add and subtract (die Strichrechnung). \( 2 + 3 \cdot 4^2 = 2 + 48 = 50 \).

Wrong would be (2 + 3) · 4^2 = 80 .
Wrong would be \( (2 + 3) \cdot 4^2 = 80 \).

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

Calculate: a) \( (-3)^2 \) b) \( -3^2 \) c) \( (-2)^3 \) d) \( 5 - 2^2 \cdot 3 \)

Show solution

a) \( 9 \) b) \( -9 \) c) \( -8 \) d) \( 5 - 12 = -7 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: powers with a negative base (with and without brackets) and the order of operations (brackets, powers, multiply/divide, add/subtract). 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.