Proof that (3)^{0} = 1
How to prove that a number to the zero power is one
Why is (3)^{0} = 1? How is that proved?
Just like in the lesson about negative and zero exponents, you can look at the following sequence and ask what logically would come next:

You can present the same pattern for other numbers, too. Once your child discovers the rule for this sequence is division by 3 at each step, then the next logical step is that (3)^{0} = 1.
The video below shows this same idea: teaching zero exponent starting with a pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof. The video also shows the idea for proof, explained below: we can multiply powers of the same base, and conclude from that what a number to zeroth power must be.
Zero Exponent video
The other proof idea is to first notice the following rule about multiplication (n is any integer):

Mathematics is logical and the mathematical rules work in (nearly) all cases. You have noticed how many times a rule is stated to apply 'for any integer n' or for 'all whole numbers'. That makes mathematics beautiful. So suppose we don't know what (3)^{0} is. Whatever (3)^{0} is, if it obeys this rule above, then
(3)^{7} × (3)^{0} = (3)^{7 + 0}
In other words, (3)^{7} × (3)^{0} = (3)^{7} 
(3)^{3} × (3)^{0} = (3)^{3 + 0}
In other words, (3)^{3} × (3)^{0} = (3)^{3} 
(3)^{15} × (3)^{0} = (3)^{15 +
0}
In other words, (3)^{15} × (3)^{0} = (3)^{15} 
...and so on for all kinds of possible exponents.
( In fact, we could write that (3)^{x} × (3)^{0} = (3)^{x}, where x is any whole number. That is the way you see it written in some math books, because whenever something is true for all kinds of numbers, it is written using a letter to symbolize those 'all kinds of numbers' .)
Since we are supposing that we don't yet know what (3)^{0} is, let's substitute P for it. Now look at the equations we found above. Knowing what you know about properties of multiplication, what kind of number can P be?
(3)^{7} × P = (3)^{7}  (3)^{3} × P = (3)^{3}  (3)^{15} × P = (3)^{15} 
So... What is the only number that when you multiply by it, nothing changes?
Question. What is the difference between 1 to the zero power and (1) to the zero power? Will the answer = 1 for both? Example 1: 1^{0} = ____ Example 2: (1)^{0} = ___ Answer: As already explained, the answer to (1)^{0} is 1 since we are raising the number Another example: in the expression (3)^{2} , the first negative sign means you need to take the opposite number of whatever the rest of the expression comes to. So since (3)^{2} = 9, then (3)^{2} = 9. 
Question. Why is zero with a zero expoent come up with an error?? Please explain why it doesn't exist. In other words, what is 0^{0}? Answer:Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.
See also these articles from Dr. Math: n^0 Power = 1: Defined or Proved? 
What is the difference between power and the exponent?
Varthan Power is the whole thing, or the answer. Exponent is the little number. For example, 8 is a power (of 2) since 2^{3} = 8. 3 is the exponent. See also Powers and exponents at FactMonster.com. 