Now I know this is one known fact that students just accept. Why is it that anything raised to the power of 0 is 1? We know that $2^{3}=2 \times 2 \times 2 = 8$, meaning we take the 2 and multiply it three times. So wouldn’t $2^0$ mean we multiply 2 zeroth times, and get back 2? So thats baffling.

This can be solved intuitively, like how we looked at $0!$ previously. We can look at $b^a$ as the number of ways to arrange a set of $a$ number with numbers from $1$ to $b$ numbers.

Consider $2^3$. This means we arrange the set of 3 numbers where the numbers are just 1 and 2. So we have $\{1, 1, 1 \}, \{1, 1, 2 \}, \{1, 2, 2 \}, \{2, 2, 2 \}, \{1, 2, 1 \}, \{2, 2, 1 \}, \{2, 1, 1 \}, \{2, 1, 2 \}$

And we find the number of ways to be eight!

### One Comment

• […] #1 12. Common Pitfalls in A’levels Math #2 13. Common Pitfalls in A’levels Math #3 14. Why is anything to the power of 0 always 1? 15. Classical Mathematical Fallacies #1 16. Classical Mathematical Fallacies #2 17. Classical […]