Examine the term 5^5. This means five times itself five times: 5 x 5 x 5 x 5 x 5 = 3,125. Another way to look at this is to understand that there are five groups of five items for a total of 3,125 items.
Examine the term 5^4, which means: 5 x 5 x 5 x 5 = 625. Notice that 3,125 divided by 5 equals 625.
Examine the term 5^3: 5 x 5 x 5 = 125. Again, notice the pattern as you work your way down the exponential amounts.
Examine the term 5^2: 5 x 5 = 25 and 125 ÷ 5 = 25.
Examine the term 5^1. This does not mean five times itself, i.e., 5 x 5, instead it means simply that there is one group of five. So 5^1 = 5.
Examine the term 5^0. Given the pattern shown throughout the examples, can you predict the solution? If the larger product divided by the term equals the smaller consecutive product, what does the product from the previous step equal when divided by the term? 5 ÷ 5 = 1. Therefore, 5^0 = 1.
Examine the term 0^0, zero to the zero power. For this expression, there is no solution. For one thing, any number times zero is zero. But also notice that in the previous steps, we worked backwards using division to come to the rational explanation that any number raised to zero is one. However, in this case, that would require us to divide by zero at each step, which is not possible; therefore, the answer is undetermined.