Laws of exponents for real numbers
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We already know a good bit about exponents. For example, we know if we took the number 4 and raised it to the third power, this is equivalent to taking three fours and multiplying them. Or you can also view it as starting with a 1, and then multiplying the 1 by 4, or multiplying that by 4, three times. But either way, this is going to result in 4 times 4 is 16, times 4 is 64. We also know a little bit about negative exponents. So for example, if I were take 4 to the negative 3 power, we know this negative tells us to take the reciprocal 1/4 to the third. And we already know 4 to the third is 64, so this is going to be 1/64. Now let's think about fractional exponents. So we're going to think about what is 4 to the 1/2 power. And I encourage you to pause the video and at least take a guess about what you think this is. So, the mathematical convention here, the mathematical definition that most people use, or in fact that all people use here, is that 4 to the 1/2 power is the exact same thing as the square root of 4. And we'll talk in the future about why this is, and the reason why this is defined this way, is it has all sorts of neat and elegant properties when you start manipulating the actual exponents. But what is the square root of 4, especially the principal root, mean? Well that means, well, what is a number that if I were to multiply it by itself, or if I were to have two of those numbers and I were to multiply them, times each other, that same number, I'm going to get 4? Well, what times itself is equal to 4? Well that's of course equal to 2. And just to get a sense of why this starts to work out, well remember, we could have also written that 4 is equal to 2 squared. So you're starting to see something interesting. 4 to the 1/2 is equal to 2, 2 squared is equal to 4. So let's get a couple more examples of this, just so you make sure you get what's going on. And I encourage you to pause it as much as necessary and try to figure it out yourself. So based on what I just told you, what do you think 9 to the 1/2 power is going to be? Well, that's just the square root of 9. The principal root of 9, that's just going to be equal to 3. And likewise, we could've also said that 3 squared is, or let me write it this way, that 9 is equal to 3 squared. These are both true statements. Let's do one more like this. What is 25 to the 1/2 going to be? Well, this is just going to be 5. 5 times 5 is 25. Or you could say, 25 is equal to 5 squared. Now, let's think about what happens when you take something to the 1/3 power. So let's imagine taking 8 to the 1/3 power. So the definition here is that taking something to the 1/3 power is the same thing as taking the cube root of that number. And the cube root is just saying, well what number, if I had three of that number, and I multiply them, that I'm going to get 8. So something, times something, times something, is 8. Well, we already know that 8 is equal to 2 to the third power. So the cube root of 8, or 8 to the 1/3, is just going to be equal to 2. This says hey, give me the number that if I say that number, times that number, times that number, I'm going to get 8. Well, that number is 2 because 2 to the third power is 8. Do a few more examples of that. What is 64 to the 1/3 power? Well, we already know that 4 times 4 times 4 is 64. So this is going to be 4. And we already wrote over here that 64 is the same thing as 4 to the third. I think you're starting to see a little bit of a pattern here, a little bit of symmetry here. And we can extend this idea to arbitrary rational exponents. So what happens if I were to raise-- let's say I had, let me think of a good number here-- so let's say I have 32. I have the number 32, and I raise it to the 1/5 power. So this says hey, give me the number that if I were to multiply that number, or I were to repeatedly multiply that number five times, what is that, I would get 32. Well, 32 is the same thing as 2 times 2 times 2 times 2 times 2. So 2 is that number, that if I were to multiply it five times, then I'm going to get 32. So this right over here is 2, or another way of saying this kind of same statement about the world is that 32 is equal to 2 to the fifth power.