# introducing 0th power

Let $a$ be a number not equal to zero. Then for all $n\in \mathbb{N}$, we have that ${a}^{n}$ is the product of $a$ with itself $n$ . Using the fact that the integer 1 is a multiplicative identity^{}, ($a\cdot 1=a$ for any $a$), we can write

$${a}^{n}\cdot 1={a}^{n}={a}^{n+0}={a}^{n}\cdot {a}^{0},$$ |

where we have used the properties of exponents under multiplication. Now, after canceling a factor of ${a}^{n}$ from both sides of the above equation, we derive that ${a}^{0}=1$ for any non-zero number.

Title | introducing 0th power |
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Canonical name | Introducing0thPower |

Date of creation | 2013-03-22 13:24:20 |

Last modified on | 2013-03-22 13:24:20 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Topic |

Classification | msc 00A05 |

Related topic | EmptyProduct |