Since the numerator and the denominator are both the same this becomes 1. Basically, if you have aa this equals 1. You can raise this to any power you want ,. In mathematics, you try to make everything as consistent as possible. If a function works for positive numbers , how might it work for negative numbers? For zero?
Let's extend the exponent function from the positives down to 0. It appears that next number should be 1, since that each number is the previous one divided by 2. If we try this for more numbers, we get the same pattern. 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 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'. For example, 2 3 is 8. 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 3 0 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 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!
One rule for exponents is that exponents add when you have the same base. Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:. If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:. Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.
In using exponents WHY is 0 as an exponent to a number 5 to the 0 power 1? Add comment. 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. Here we just took this, and we're just multiplying it by 1, so you're still going to get negative 8. And this might be a slightly more useful idea to get an intuition for exponents, especially when you start taking things to the 1 or 0 power.
So let's think about that a little bit. What is positive 2 to the-- based on this definition-- to the 0 power going to be equal to?
Well, we just said. This says how many times are going to multiply 1 times this number? So this literally says, I'm going to take a 1, and I'm going to multiply by 2 zero times. Well, if I want to multiply it by 2 zero times, that means I'm just left with the 1. So 2 to the zero power is going to be equal to 1.
And, actually, any non-zero number to the 0 power is 1 by that same rationale. And I'll make another video that will also give a little bit more intuition on there. That might seem very counterintuitive, but it's based on one way of thinking about it is thinking of an exponent as this.
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