my teacher asked me why a number to the zero power is equal to 1. could you please explain why a number to the zero power equals 1?

Consider first positive powers. Let's define:

f(a,n) = a^n

where a is a real number and n is a positive integer. This is well defined:

f(a,n) = a^n =
a*a*a...(n-factors in total).

Now, we haven't defined f(a,n) when n is not a positive integer, so, in theory, you are free to extend the function f(a,n) in any arbitrary way to other numbers. However, the function
f(a,n) has some nice properties and you want to preserve those when you extend the definition of f(a,n).

E.g.:

f(a,n+m) = f(a,n)*f(a,m)

If we want to extend the function to
n = 0 while not violating this equation, then we must choose f(a,0)= 1. To see this, take m = 0, in the above equation:

f(a,n) = f(a,n)*f(a,0) --->

f(a,0) = 1.

So, what mathematicians have done here (and in many other cases) is to take some of the properties of the function they want to extend to a larger set as the definition of the function (they uniquely define the function).

thx but this didn't help i need to know why n^0= 1

He told you, and I think it was the coolest answer yet!

Of course! To understand why a number to the zero power equals 1, we can use the concept of exponents.

When you raise a number to a power, you are essentially multiplying that number by itself a certain number of times. For example, 2 raised to the power of 3 (2³) means multiplying 2 by itself three times: 2 x 2 x 2 = 8.

Now, let's consider what happens when we lower the exponent. If we divide 2³ by 2², we get 8 ÷ 4 = 2. Similarly, if we divide 2² by 2¹, we get 4 ÷ 2 = 2. Continuing this pattern, if we divide 2¹ by 2⁰, we get 2 ÷ 1 = 2.

This pattern suggests that when we divide a number by itself, we always get 1. So, if we divide 2⁰ by 2⁰, we should get 1. However, 2⁰ is actually equal to 1, so dividing 1 by 1 will always give us 1. This is why any number raised to the power of 0 is equal to 1.

In summary, raising a number to the power of 0 represents dividing a number by itself, which always gives us 1. Hence, any number raised to the power of 0 equals 1.