Why exponent to power of 0 is one

Lets call any number x:. Raising something to a power greater than zero means multiplying it by itself a number of times equal to the power. So, for instance,. 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.

Now, also note that if you raise something to a negative power, then you take the reciprocal of that something:. 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. Why any number to the zero power always gives a one? Answer 1: This is an excellent question! Answer 2: It's exciting to me that you asked this question. 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 a b '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'.

Since 1 is the multiplication identity, the product is never changed by multiplying by 1. When we multiply numbers, we can start with a 1 identity and then multiply it by the rest of the numbers, just like when we add or subtract numbers, we can start with 0 identity , and then move to the right addition or left subtraction on the number line.

Now consider that an exponent power is merely the number of times you need to multiply the base the five ; and don't forget to also multiply by one 1. Now, move the 5 in the denominator to the numerator. Sign of expressions challenge problems. Practice: Signs of expressions challenge. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript What I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples.

So in the last video, we saw that taking something to an exponent means multiplying that number that many times. So if I had the number negative 2 and I want to raise it to the third power, this literally means taking three negative 2's, so negative 2, negative 2, and negative 2, and then multiplying them.

So what's this going to be? Well, let's see. Negative 2 times negative 2 is positive 4, and then positive 4 times negative 2 is negative 8.

So this would be equal to negative 8. Now, another way of thinking about exponents, instead of saying you're just taking three negative 2's and multiplying them, and this is a completely reasonable way of viewing it, you could also view it as this is a number of times you're going to multiply this number times 1. So you could completely view this as being equal to-- so you're going to start with a 1, and you're going to multiply 1 times negative 2 three times.

So this is times negative 2 times negative 2 times negative 2. So clearly these are the same number. Mathematicians eventually decided that such an argument was undecidable, so they decided to leave 00 undefined.

Pre-Algebra Laws of Exponents. Explanations 3 Caroline K. Zero Exponent Rule. Image by Caroline Kulczycky. In other words, if you raise a nonzero number to the power of 0, the result is 1.

Mathematicians debate the value of Some say it's 1, and some say it's undefined. Related Lessons.