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 Introducing0thPower 2013-03-22 13:24:20 2013-03-22 13:24:20 mathcam (2727) mathcam (2727) 8 mathcam (2727) Topic msc 00A05 EmptyProduct