 For the integers if you go one level above the rational numbers are one One level above the whole numbers you get integers integers do negative numbers, okay, so Well they introduce negative whole numbers So when you have your natural numbers here we went up to the whole numbers whole numbers introduce zero anything to the power Zero is one now if you put a negative number in the little box up there What it does to the big box it flips it so If you had five in the big box and negative three here the way it work If the way what the negative number would do It would put it one over five to the power of three So you use up the negative to flip the number five Okay, and if the number if the base number if this number is in the bottom to a negative power What it does it kicks it up if it's a fraction in the in the big box What it does it flips it the bottom goes up top and the top goes to the bottom if you have a negative exponent Okay, so let's talk about integers negative numbers negative numbers eight to the power of negative four And negative power means if the number is on top you kick it down. That's the base So what this means is this goes down So if you had This would be one over a square Now it also works the other way around so if it's on the top That's a negative power kicks it down if it's in the bottom and if it's a negative power kicks it up. So if you had two over a to the negative three Well, the negative power down here kicks the a on top. So this big box to the power to a cube now this cube only applies to the a because There is no bracket here saying it applies to both of them. So keep your Keep track of your variables and your numbers. Don't do operations to numbers that they're not supposed to be done to This also applies to a fraction So if it's up top if it's a negative power kicks it down if it's in the bottom negative power kicks it up Well, if it's a fraction to a negative power, it just flips it. So if you had a over To the negative two, okay, all that does is Deal with the negative number It takes the a kicks it down. It takes the b kicks it up. So this would be the over a After the integers we had Rational numbers Now rational numbers are basically fractions any number that repeats or terminates is a fraction Okay, so the way it works with fractions is if you have two over three the denominator in the exponent becomes a radical becomes the root so if Okay, let's take a green. I'm gonna do this thing right down here Okay, so if you have a if you have any any number in the base and put it to a fraction What happens is the top number in the fraction is the numerator thing That's what they call it that acts like the natural numbers or the whole number Well, you wouldn't put zero in the fraction that acts like natural numbers So the top number just clones whatever is in the base. Okay, the denominator in the exponent goes to the root So for example, I'm gonna do Five to the power of two over three. Let's do that. We got five to the power of Two over three. The way it works is that three in the denominator moves To the to the radical Okay, awesome So the way it works is we got a skater who wants to skate. He's been nice taking a break so what we got is five to the power of two over three and Grabs the three in the denominator and Moves it to the front of the five and it becomes the radical So that's the way Rational numbers work says Eight to the power of Three and if you want to write this in a radical form, this would be you take three you move it up front Under this root symbol. Okay, so this would be a cute root Makes sense. What if you had you could go the other way around as well What if you had to the power of ten? Well the way this works is this number here Goes in the denominator in the power So this To the power of and ten divided by five is just two so the answer to this would be to rule This property helps you to reduce to evaluate things simplify things Let's do a few more. So this really sits in if I said hey, what's your form? To a power of a half. Well, we talked about it and you should know right now that The two here goes in the root symbol So this is just a square root of for Now what is the square root of for me? You should have already told me that square root of four is two, right? the way it works is This root symbol here is a boundary and the way it works is the number here Tells you what's required to go past this boundary either this way or this way Right now. This is a root. That was a two. So this just means square root of four square root of four You break it down into its prime factors So what the radical here means is for you to go from the inside out You need two things that are the same and two things combined to become one thing so you pair these guys up these guys come outside of the root symbol and Two roots coming out. There's nothing else left in there. So square root of four becomes This applies to any number that's in the root. So for example actually, let's do one more square root first Let's say if I said Well eight to a power of half is square root of eight Right square root of eight is Let's see what it is Now as we said this radical symbol here this root symbol because that's a two There's no other number here. It means you're looking for pairs So you start pairing up your numbers So two twos can come out of the symbol move this way as a single so they join together So two twos become a two and what's left inside the root symbol inside the root symbol? You still have a two left right down there So eight to a power of half is a square root of eight a square root of eight becomes two root two Now that if you remember is an irrational number because it's the root of a prime number that means the number there doesn't Doesn't end or repeat. So it keeps on going changes up. No pattern. So this applies to any Radical so for example, what if you had what's the cube root? What's the cube? Eight well as a power that would have been So that would be the cube root of eight well eight you can break down crime numbers charge two times two times two For in this case, this is a cube root. So what that means is you're looking for triplets Triplets so you combine three things that are the same Three twos here. So the rule is the boundary or the toll for you to go from the inside out You need triplets triplets become one. So these three twos come out of this And you become a two now you took everything out of the root symbol The cube root symbol so there's nothing left. So the cube root of eight or eight to a power of a third is a cube root of eight Which is just two then you have 27 Now if you remember if you want to crunch numbers reduce numbers Are simplified things you want to reduce the numbers first instead of making it bigger, right? Remember when we're doing multiplying fractions, you didn't multiply all the numbers on top And then breaking the prime numbers kill them right what you did was kill whatever you could kill and then multiply things So the same type of principle works here when you're evaluating numbers simplifying numbers What you want to do is reduce them first and then make them bigger if you have