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 the rule for this sequence is division by -3 at each step, then the next logical step is that (-3)⁰ = 1.

The other proof idea 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:

n^x × n^y = (n×n×n ×...×n×n×n) × (n×...×n) = n^{x + y}

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: -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 in that case 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 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 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 think of 0⁰ also 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 logs wouldn't work with it.

So because of these kind of problems, it is often left "indeterminate".

However, if zero to zeroth power needs to be defined as having some value, 1 is the most logical definition for its value. This can be "handy" if it makes some law to work (such as binomial theorem, or in combinatorial mathematics).

See also these articles from Dr. Math:

n^0 Power = 1: Defined or Proved?
Why are Operations of Zero so Strange?
What is 0 to the 0 power?

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 2³ = 8. 3 is the exponent. See also Powers and exponents at FactMonster.com.

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Proof that (-3)0 = 1 How to prove that a number to the zero power is one

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Proof that (-3)⁰ = 1
How to prove that a number to the zero power is one