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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:

(-3)4 = 81
(-3)3 = -27
(-3)2 = 9
(-3)1 = -3
(-3)0 = ????

You can present the same pattern for other numbers, too. Once your child discovers that the rule for this sequence is that at each step, you divide by -3, 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.

The other idea for a proof is to first notice the following rule about multiplication (n is any integer):

n3 · n4 = (n·n·n) · (n·n·n·n) = n7

n6 · n2 = (n·n·n·n·n·n) · (n·n) = n8

Can you notice the shortcut?  For any whole number exponents x and y you can just add the exponents:

nx · ny = (n·n·n ·...·n·n·n) · (n·...·n) = nx + y

Mathematics is logical and its rules work in all cases (theorems are stated to apply "for any integer n" or for "all whole numbers"). So suppose we don't know what (-3)0 is. Whatever (-3)0 is, if it obeys the 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 can write that (-3)x · (-3)0 = (-3)x, where x is any whole number.

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

In other words... 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 be 1 for both?

Example 1: -10 = ____
Example 2: (-1)0 = ___

Answer: As already explained, the answer to (-1)0 is 1 since we are raising the number -1 (negative 1) to the power zero. However, in the case of -10, the negative sign does not signify the number negative one, but instead signifies the opposite number of what follows. So we first calculate 10, and then take the opposite of that, which would result in -1.

Another example: in the expression -(-3)2, the first negative sign means you take the opposite of the rest of the expression. So since (-3)2 = 9, then -(-3)2 = -9.

Question. Why does zero with a zero exponent come up with an error?? Please explain why it doesn't exist. In other words, what is 00?

Answer: Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.

Since x0 is 1 for all numbers x other than 0, it would be logical to define that 00 = 1.

But we could also think of 00 having the value 0, because zero to any power (other than the zero power) is zero.

Also, the logarithm of 00 would be 0 · infinity, which is in itself an indeterminate form. So laws of logarithms wouldn't work with it.

So because of these problems, zero to zeroth power is usually said to be indeterminate.

However, if zero to zeroth power needs to be defined to have some value, 1 is the most logical definition for its value. This can be "handy" if you need some result to work in all cases (such as the binomial theorem).

See also What is 0 to the 0 power? from Dr. Math.

What is the difference between power and the exponent?

The exponent is the little elevated number. "A power" is the whole thing: a base number raised to some exponent — or the value (answer) you get if you calculate a number raised to some exponent. For example, 8 is a power (of 2) since 23 = 8. In this case, 3 is the exponent, and 23 (the entire expression) is a power.

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