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Exponent zero and negative exponents

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

What is \( 2^0 \)? What is \( 2^{-3} \)? The laws of exponents (die Potenzgesetze) should still be true. That is how we find: \( a^0 = 1 \) and \( a^{-n} = \frac{1}{a^n} \).
📄 Open worksheet (PDF, German)

The video

Watch the colours: the base is blue, 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. 0:21 Exponent zero
  3. 0:59 The rule for zero
  4. 1:34 Negative exponents
  5. 2:20 A pattern
  6. 2:48 The rule for minus
  7. 3:20 Not negative!
  8. 3:37 A look ahead
  9. 3:53 Try it yourself

The most important points

Writing the two zero times does not work. Our idea: the laws of exponents should still be true. \( 2^3 = 2^{3+0} = 2^3 \cdot 2^0 \), so \( 8 = 8 \cdot 2^0 \). Multiplying by \( 2^0 \) changes nothing. So \( 2^0 = 1 \).

The laws of exponents should still be true. That is how we find 2^0 = 1 .
The laws of exponents should still be true. That is how we find \( 2^0 = 1 \).

Exponent zero

Every number to the power of zero is one: \( a^0 = 1 \) (for \( a \neq 0 \)). Check: \( 2^3 : 2^3 = 8 : 8 = 1 \) and \( 2^3 : 2^3 = 2^{3-3} = 2^0 \). Careful: \( 2^0 \) is not \( 0 \) and not \( 2 \).

\( 2 = 2^1 = 2^{4-3} = 2^4 \cdot 2^{-3} = 16 \cdot 2^{-3} \). Because \( 16 \cdot \frac{1}{8} = 2 \), \( 2^{-3} = \frac{1}{8} = \frac{1}{2^3} \).

When the exponent gets one smaller, we divide by 2: 16, 8, 4, 2, 1, ½, ¼, ⅛.
When the exponent gets one smaller, we divide by 2: 16, 8, 4, 2, 1, ½, ¼, ⅛.

Negative exponent

A minus in the exponent turns the power into a fraction (der Bruch): \( a^{-n} = \frac{1}{a^n} \). Especially important: \( a^{-1} = \frac{1}{a} \) is the reciprocal (der Kehrwert) of \( a \). Examples: \( 5^{-1} = \frac{1}{5} \), \( 3^{-2} = \frac{1}{9} \), \( 10^{-2} = \frac{1}{100} \).

A negative exponent makes a fraction, not a negative number: 2^{-3} = 1/8 , not -8 .
A negative exponent makes a fraction, not a negative number: \( 2^{-3} = \frac{1}{8} \), not \( -8 \).

A look ahead: a fraction can also be an exponent, for example \( 4^{\frac{1}{2}} \). We will learn this later with roots (die Wurzel).

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)

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

Task

Calculate: a) \( 7^0 \) b) \( 4^{-2} \) c) \( 10^{-3} \) d) \( 2^3 \cdot 2^{-3} \)

Show solution

a) \( 1 \) b) \( \frac{1}{16} \) c) \( \frac{1}{1000} \) d) \( 2^0 = 1 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: exponent zero and negative exponents (a to the 0 = 1, a to the minus n = 1 divided by a to the n). 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.