Why is anything to the 0th power 1?

Question

Its multiplying by itself zero times, so shouldnt it be 0?

Answer 1

Here's another way you can think about it:

Here we have a pattern:

3^3 = 27

3^2 = 9

3^1 = 3

To get from 27 to 9 we divided 27 by 3, to get from 9 to 3 we divided 9 by 3, so in order to follow the pattern we should also divide 3 by 3 to get to the next number. Which would be 1. So looking at the entire pattern again we have:

3^3 = 27

3^2 = 9

3^1 = 3

3^0 = 1

Having 3^0 = 0 wouldn't make sense because we wouldn't be able to get to 3 if we were to multiply 3^0 by 3. 1 is sometimes called the Multiplicative Identity for reasons related to this topic. You can kind of think of 1 as the 0 of multiplication.

As was mentioned raising 0 to the 0 power is NOT 1 and is the only exception to the rule that anything to the 0 power is 1. The reason it is undefined is for kind of the same reason 3^0 = 1. Let's look at the pattern like we did for 3:

0^3 = 0 because 0 * 0 * 0 = 0

0^2 = 0 because 0 * 0 = 0

0^1 = 0 because 0 = 0

but we cannot use the same logic as before with 3 because it is a rule that you can never divide by 0. So for that reason 0^0 is undefined.

I hope that helps.

Answer 2

No, here is one way you can think of it:

 

Remember the rule that:

 

x^m/xⁿ = x^m-n   (if x≠0)

 

So if m = n, this says:

 

x^m/x^m = x⁰

 

But clearly x^m/x^m = 1, so we must have that     x⁰ = 1 as well.

 

Be careful when you say that "anything" to the 0th power is 1, because 0⁰ is undefined.

Answer 3

To understand this concept, let’s review the rules of exponents for multipling and dividng exponents.

The rules of exponents states that x^m/x^{n =} x^m-n

x^{0 power is the  smallest power for any number.}

Let's  use the powers of 7 as an expample.

7⁵/7 ⁼  7^5-1= 7⁴      =16807/7=2401

7⁴/7= 7^4-1=  7³     =2401/7=243

7³/7 = 7^3-1 = 7²      = 343/7 = 49

7²/7= 7^2-1 = 7¹      = 49/7 = 7

7¹/7 = 7^1-1 = 7⁰      = 7/7= 1

Applying this law explains why any number (with the exception of 0) to the zero power is always equal to 1.

Answer 4

Ben's answer is a good description!

Here is something that is NOT so much a mathematical description, but just a way to help remember it:
Think of it as x^n = (1) times x, n times.

This would also work for negative exponents.
x^(-n) = 1 / x^n
Think of x^(-n) as (1) divided by x, n times.

Answer 5

Another way you may consider if you have been introduced to fractional powers:

You might remember that a^1/n is the number that must be multiplied as n copies to get a.  In other words, (a^1/n)ⁿ = a.  Consider the following set of equations:

(16¹)¹ = (16.00)¹ = 16

(16^1/2)² = (4.00)² = 16

(16^1/3)³ = (2.52)³ = 16

(16^1/4)⁴ = (2.00)⁴= 16

. . . etc.

As the power attached directly to the orange 16 becomes arbitrarily small (i.e. 16^1/100, 16^1/1000, 16^1/10000, and so on), we obtain quantities closer and closer to 1.00.  They have to be very close to 1, or else we would quickly "overshoot" the final value of 16 before we finished multiplying together all our copies of orange numbers.  So, as the power attached to the 16 becomes arbitrarily close to zero, the orange quantity becomes arbitrarily close to one, and when the power equals zero, the quantity equals one.  

Can you see why the quantity gets close to one and not to zero?  Suppose that 16^1/000 were not close to 1, but, instead, a small number close to zero, like 0.01.  If this were true, then (16^1/1000)¹⁰⁰⁰ = (0.01)¹⁰⁰⁰, which is a very small number, much smaller than one (0.01 multiplied by itself many, many times is very small).  However, this cannot be true because (16^1/1000)¹⁰⁰⁰ = 16, which is not such a very tiny number.  Thus, we would not intuitively expect 16⁰ to be zero.  

 

Hope this was interesting.  

Answer 6

You are not multiplying 3 times zero, you are just not multiplying by itself at all. The number 3 is not being multiplied so it remains the same.