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Why is any nonzero number raised to the power of zero equal to one?
Answer 1:

Let's first agree on some conventions. For example, that some number a raised to the power of N, where N is an integer (such as 3, 105, -23) meansa * a * a * a multiplied N times.

So a1 = a
and
a2 = a * a
or if N is negative that it is divided N times, so
a(-2) = (1/a) * (1/a) = 1 / (a2)
In this case, (a2 ) / (a2) = (a * a) / (a * a) = 1

But when it comes to division, we also know that
aN / aM = a (N-M)

For example, a5 / a3 = ( a * a * a * a * a ) / (a * a * a) = a * a = a2 = a(5-3)

So, back to our previous example, a2 / a2 = a(2-2) = a0 but also,
a2 / a2 =
(a * a )/(a * a) = 1
Thus, a0 = 1

So for example, when you go from 24 to smaller powers, every time you divide by 2.

24 = 16
23 = Divide by 2 the 24 = 8
22 = Divide by 2 the 23 = 4
21 = Divide by 2 the 22 = 2
20 = ??? Divide by 2 the 21 = 1
2(-1) = Divide by 2 the 20 = 1/2 = 0.5

Hope these examples help.

Answer 2:

A simple example to show this is that if you go from 21 to 22 (2 to 4), you multiply by 2.

The same is true if you go from 22 to 23 (4 to 8).

To go in the other direction, you would divide by 2. So to go from 23 to 22, you would divide by 2. So if you want to go to 21 to 20, you would also divide by 2. If you divide 2 by 2, you get 1. The same is true of any number, to get to the zero power, you divide the number by itself which gives you 1.


Answer 3:

This is a good question. If you take any number and divide it by itself, the answer you get will be 1.

For example:
3/3 = 1 or 9/9 = 1

This applies to *all* numbers, even numbers raised to a power. For example:
32/32 = 1

This makes sense because 32 is just 3*3, which of course is just 9.

Now, one of the rules of powers says that as long as the base number is the same, all you have to do to divide two numbers raised to a power is to just subtract the top power from the bottom power,. like this:

34/32 = 3(4-2) = 32 = 9

This makes sense because 34 is really just 3*3*3*3, and 32 is just 3*3, so the question is really just asking:

(3*3*3*3)/(3*3) = ?

And we know from arithmetic rules that we can cancel same numbers on the top and bottom of the division and get:

34/32 = (3*3*3*3)/(3*3) = 3*3 = 9

But notice that this rule only works when the base number (3 in this case) is the same for both the top and bottom.

What happens when the power on both the top and bottom number is the same? For example:

32/32

We know from earlier that the answer will be 1, which is a lot like the answer you get when you raise any number to the power of zero, isn’t it? Let’s apply the rule of powers from before:

32/32 = 3(2-2) = 30

And we already know that the answer to 32/32 is 1, so we can see that
30 = 1

This is exactly what we set out to prove! That is because
30 = 32/32 = 1

Now you can try this same exercise with any number at all and you will get the same answer.


Answer 4:

It's because of the way that negative powers work. Think about this:

21 = 2,
22 = 4,
2-1 = 1/2,
and 2-2 = 1/4.

20 is no multiplication or division by 2 whatsoever. The only number that you could multiply by 2n and actually get 2n is 1. Therefore, 20 = 1.

This same reasoning works for any other non-zero number. Try it with 3 and see for yourself!


Answer 5:

Any time we use powers, we can rewrite them as division, like this: n(x-y) = nx/ny.

For example, 32 = 33-1
= 33/31
= 27/3 = 9.

If we apply this to the case of a zero power, we can rewrite:
30 = 31-1
= 31/31 = 1!

We can make this more general and say that any number (n) with power of zero is equal to n/n = 1.


Answer 6:

To answer this, we need to know some of the properties that powers have. First, a raised to the b power is equal to a multiplied by itself b times. For a = 2 and b = 4, 24 = 2 x 2 x 2 x 2 = 16. This is straightforward for a positive b. What about a negative b?

A negative power is the reciprocal (or fraction of one) of the positive value. For example, if b = -4 in the example above: 2-4 = 1/(2 x 2 x 2 x 2) = 1/16

Second, we need to know that powers can be added together if the base (a) is the same. As examples, 23 x 21 = 2(3+1) = (2 x 2 x 2) x 2 = 24 = 16.

Now, we have what we need to explain why a base (non-zero) to the zero-th power must be one, even if imagining multiplying something zero is confusing. Let’s go to our initial example: 24 can also be expressed as 24 x 20 because it is also equal to 2(4+0) = (2 x 2 x 2 x 2) = 24.

The only number that can multiply a non-zero number without changing its value is one.

Another way to think about it is to start with 20, which is equal to 2(4-4) = 24 x 2-4. From before, we know that 24 x 2-4 = 16 x (1/16) = 1. This is true for any nonzero number.

Things are tricky when you try to think about zero to the zeroth power – it is considered “undefined”, not a number at all.



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