UCSB Science Line
 Why any number to the zero power always gives a one? Question Date: 2011-09-01 Answer 1:This is an excellent question! There are lots of different ways to think about it, but here's one: let's go back and think about what a power means. When we raise a number to the nth power, that really means that we multiply that number by itself n times, so for example, 22 = 2*2 = 4, 23 = 2*2*2 = 8, 34 = 3*3*3*3 = 81, and so on. So when we raise a number to the zeroth power, that means we multiply the number by itself zero times - but that means we're not multiplying anything at all! What does that mean? Well, let's go even farther back to the simplest case: addition. What happens when we add no numbers at all? Well, we'd expect to get zero, because we're not adding anything at all. But zero is a very special number in addition: it's called the additive identity, because it's the only number which you can add to any other number and leave the other number the same. In short, 0 is the only number such that for any number x, x + 0 = x. So, by this reasoning, it makes sense that if adding no numbers at all gives back the additive identity, multiplying no numbers at all should give the multiplicative identity. Now, what's the multiplicative identity? Well, it's the only number which can be multiplied by any other number without changing that other number. In short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1. Answer 2:It's exciting to me that you asked this question. The fact is these rules are presented as somewhat arbitrary, but there is always, always (well almost always) good reason for them. Keep it up! If it ever sounds arbitrary then hound your teacher. If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question: Mathematics was initially developed to describe relationships between everyday quantities (generally whole numbers) so the best way to think about powers like ab ('a' raised to the 'b' power) is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'. For example, 23 is 8. Why? There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2:(1,1,1) (1,1,2) (1,2,1) (2,1,1) (2,1,2) (2,2,2) (1,2,2) (2,2,1). So what does 30 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table? Just one... don't put anything on the table and that's your only option. Therefore it's consistent to say 30 = 1. There are other reasons why a0 has to be 1 - for example, you may have heard the power rule: a(b+c) = ab * ac. What happens if b = 0? Well I know that a(0+c) is the same as ac and by the first formula this is also a0 * ac. What choice for a0 make sense to satisfy ac = a0 * ac? Clearly a0 = 1. I know this sounds a little fishy since we started with a rule I could have just made up (which is why I gave the other reason first), but these formulas are all consistent and there is never any magic step, I promise! You can always work backwards to prove a(b+c) = ab * ac etc etc. Hope this helps! Answer 3:Any number to the zero power always gives one.One rule for exponents is that exponents add when you have the same base. So if you have a number, x, and exponents, a and b, then:xa * xb = x(a+b)So then if we make one of the exponents negative:xa * x-b = x(a-b)And if the exponents are the same magnitude (a = b)xa * x-b = xa * x-a = x(a-a) = x0Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:x-a = 1/xaSo, we can also write xa * x-a in a different way:xa * x-a = xa * 1/xa = xa/xaAnd a number divided by itself is always 1 so:xa * x-a = xa* 1/xa = xa/xa = 1:So now we've shown that:xa * x-a = x(a-a) = x0andxa * x-a = xa * 1/xa:This means that any number x0 = 1.If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:If we plug in numbers, (for example let x = 5, a = 2, and b = 4) then:One rule for exponents is that exponents add when you have the same base.52 * 54 = 5(2+4) = 56 = 15625So then, if we make one of the exponents negative:52 * 5-4 = 5(2-4) = 5-2 = 0:04And if the exponents are the same magnitude:52 * 5-2 = 5(2-2) = 50Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:5-2 = 1/52 = 0:04So we can also write 52 * 5-2 in a different way:52 * 5-2 = 52 * 1/52 = 52/52 = 25/25And a number divided by itself is always 1 so:52 * 5-2 = 52 * 1/52 = 52/52 = 25/25 = 1So now we've shown that:52*5-2 = 5(2-2) = 50and52 * 5-2 = 52/52 = 1This means that 50 = 1.This works for any number x that you want to plug in except for x = 0,because 0/0 is indeterminate (it is like dividing zero by zero). Answer 4:Let's look at what it means to raise a number to a certain power: it means to multiply that number by itself a certain number of times. Three to the second power is three multiplied by itself 2 times, or 3*3=9. Let's look at a few examples:35 = 3*3*3*3*3 = 24334 = 3*3*3*3 = 8133 = 3*3*3 = 2732 = 3*3 = 931 = 3 = 3But how do you go from 31 to 30? If you look at the pattern, you can see that each time we reduce the power by 1 we divide the value by 3. Using this pattern we can not only find the value of 30, we can find the value of 3 raised to a negative power! Here are some examples:30 = 3/3 = 13-1 = 1/3 = 0.3333... (this decimal repeats forever)3(-2) = 1/3/3 = 0.1111...3(-3) = 1/3/3/3 = 0.037037...No matter what number we use when it is raised to the zero power it will always be 1. Suppose instead of 3 we used some number N, where N could even be a decimal. N1=N, and to reduce the power by 1 we divide by N, soN0=N1/N = N2/2N =1.Notice that 3(-1) is the same as 1/(31), 3(-2) is the same as 1/(3(2)),and so on. This gives us a useful property of exponents, namely that a(-b) is the same as 1/(ab). Answer 5:Heres a quick demonstration of why any number (except zero) raised to the zero power must equal 1. As an example we will let that any number be the number 3.Note that:31 = 3 = 332 = 3*3 = 933 = 3*3*3 = 2734 = 3*3*3*3 = 81And so on Youll notice that 33=(34)/3, 32=(33)/3, 31=(32)/3In other words, 3(n-1)=(3n)/3So 30=(31)/3=3/3=1 This same reasoning will work for any number (not just 3), except the number 0. It wont work for 0 because you cant divide by 0. Lets call any number x: x(n-1)=xn/xSo x0 = x(1-1) = x1/x = x/x = 1 Answer 6: of the answer is that this is how we've defined powers to be. Raising something to a power greater than zero means multiplying it by itself a number of times equal to the power. So, for instance, 21 = 2 22 = 2 x 2 = 4 23 = 2 x 2 x 2 = 8 and so on. Now, you can multiply anything by 1 and it will still be the same thing, and likewise you can divide anything by 1 and it will still be the same. Therefore: 21 = 2 x 1 = 2 22 = 2 x 2 x 1 = 4 23 = 2 x 2 x 2 x 1 = 8 You see I've just multiplied everything by 1. Now, also note that if you raise something to a negative power, then you take the reciprocal of that something: 2-1 = 1/2 2-2 = 1 /(2x2) = 1/4 2-3 = 1/8 And so on. Again, we can multiply by everything by 1: 2-1 = 1 x 1/2 Now, what happens when the power is zero? Well, you're not multiplying by anything, except the 1 you started with. You're not dividing by anything, except the 1 you started with. So, what you're left over with is 1. Now, here is the slightly more mathematically sophisticated version: when you raise something to a power, what you do is take 1 and multiply it by the base of the power a number of times equal to the power. So, by definition, raising something to the power of zero means you start with 1, and then don't multiply it by anything. So, naturally, 1 is what you're left over with. Click Here to return to the search form.

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