## Intermediate Algebra (6th Edition)

When we multiply a base by an exponent, we are essentially taking the product of 1 and the base multiplied by the amount of times specified by the exponent. For example, $5^{2}=5\times5=1\times(5\times5)$ Therefore, when the base is 5 and the exponent is 2, we are multiplying 1 by 2 fives. However, when the exponent is 0, we are multiplying 1 by 0 fives. Therefore, $5^{0}=1$ and, more generally, $a^{0}=1$ (where a is a nonzero real number).
We can further understand this result by using the product rule. $a^{0}\times a^{n}=a^{0+n}=a^{n}=1\times a^{n}$ $a^{0}\times a^{n}=1\times a^{n}$ Divide both sides by $a^{n}$. $a^{0}=1$