PROPRIEDADES das POTÊNCIAS \Prof. Gis/

For those of you who have already attended that class in which everything about potentiation is explained, right from the beginning to the entire definition of potentiation, I invite you to now watch this class on the properties of potentiation, but why will the definition of potentiation differ from these properties ? ? Guys, when we work on these properties that I 'm talking about, our calculations become much simpler to carry out, so be sure to watch this one until the end and check out all these properties, shall we?

Take the opportunity to subscribe to the channel and give Gis a thumbs up, help with the class. So let's talk about the first property, not necessarily in order, because we say first property, second, third, you don't need to memorize the order of these properties, you just need to know what to do in each one, it's worth it. So this property that I brought is the product of powers with the same base, so it's very important if you didn't watch the class in which I explained about the definition of power, it's important that you watch it because otherwise you won't understand 100% of what I'm going to explain.

here it's good so it said here that it's the good product, let's try to understand what the product is, the product is related to multiplication because the product is the result of a multiplication, so remember, you saw the word product, it reminds you of multiplication, powers of the same base, so it means that the base has to be equal to the base is this number that is here and this number that is up here is called exponent, so just to give you a reminder, okay, so what do I do when I have this situation here, I'm going to keep the base, I repeat this base, I'm going to add the exponents, oh Gis, but why do you do it like that, so this first case here I'm going to do it differently for you, so here I have 5, 5 when there is no exponent there it is because it is 5 to the power of a certain number, so I have 5 to the power of 1 times 5 to the power of 2 so if I were to solve this, it would be 5, right, because 5 to the power of 1 is 5 itself and then here is 5 to the power 2 would be 5 x 5, okay. So see that this one here is from the first factor of this first power and these other two here are from 5 to the power of 2, right now how many factors 5 do we have there now we can analyze that we have one two three factors 5. So that means that the answer , we always say that when we work with the properties of powers, the statements they bring like this leave the result in the form of a power, okay, so here it would be five to the power of 3 because I have three factors here, right, because what is the potentiation people, it is a repetition, right, of equal factors, when I do a multiplication of equal factors, a multiplication of equal factors, then it resulted in 5 taking the third, so that's why it says here, in this property we must maintain the base and add the exponents so if I keep the base here which is 5 and add the exponents, one plus two would give three it would be straight forward so I would skip this process of writing all the repeated factors it is much faster so now we are going to apply everything straight power multiplication with the same base OK, all bases are 2 so I'm going to keep the base 2 I'm going to add the exponents 4 plus 2 from here and you must remember that when there's no exponent here it's because it's an exponent 1 so here plus 1 , so here I will get as an answer 2 to the power of 4 but 2 of the 6 plus 1 of the 7 so 2 to the seventh power Ok.

In this other case here with decimal how do you do it with decimal in the same way as long as the base is the same , and there the base is what is 0. 3 which is three tenths, right, so here it would be 0. 3 exponents, let's add it directly and I will add the exponent that is here that there is no one so it is 1, 1 + 5 which will be equals 6, that's it, look how simple it is to do.

In this other case, I love it when there are these negative numbers, so the students end up getting confused and then it generates that discussion in the classroom, so the base is the same, keep the base and now we're going to add the exponents, so it's going to be minus 2, I'm going to add, right, with exponent - 4. So see that here we need to do the sign rule, to remove that parentheses, so here, by doing the sign rule, I will get minus, right, because more with less will be less, so I will have 3 raised to minus 2 so here - 4 and how much Now gives negative two with negative four? So see that I have two debts so I'm going to add those two, it's going to be equal to negative six so here I'm going to have 3 to the power of negative 6 so guys, this is very important here, it generates a lot of conflict and a lot of confusion among students, this is the rule of signal here with the exponents, so it works and scores well.

In this other case here now I put a letter and how are you going to do it with the letter the same way first thing you will look at the base the base is repeat the base which is ox and I'm going to add the exponents I'm going to add them because here I have a multiplication so I'm going to do 8 plus minus 3 then apply the sign rule here which will be minus ok so my answer here will be X to the power of 8 - 3 so I will have X raised to the power of 5, the indication because then you remove the doubt and don't take that doubt when solving, so here are exercises that equal the properties of the powers. And in this case now with a fraction, so see that I'm varying a lot, right, doing several cases again, let's analyze the base, the bases are the same Ok so I should keep the base which is 2/5 and I'll add the exponents then I'll do 6 plus - 4 then it will be 6 plus - 4 and I do the sign rule here it will give minus then I will do 6 - 4 so I will get 2/5 and then I do it here folks, 6 - 4 will be equal to 2 so I got the power of 2/5 raised to 2 Okay, so take a good look at all of this about the properties of the powers, there is also a doubt that Gis always comes up and if I have an exercise that not all the bases were the same, then I can apply the property in the same way, right? I can only apply it if the bases are the same, so for example if here were a 3 and here were 2 and here were 2 then I would do it with these two here I would apply this property, but with If this other number wasn't 2, I wouldn't be able to include it, okay, it would just be there on the side, right guys?

But rest assured that in the majority, in most exercises, it always brings it back, since it is for you to apply this concept, it will bring the bases equal, okay, and now in the division of powers with the same base, you already have an idea of ​​what it does because in the product that It's in multiplication, right now in division we 're going to do here I added the exponents so in division we're going to subtract the exponents. So look at the examples I'm going to show you now. So see that here we are going to do 5 to the power of 5 divided by 5 to the power of 3 so here we have what we have a division of powers of the same base and what I am going to do we have already seen that we are going to subtract but I will show why I subtract by doing the same procedure that I did in multiplication, so 5 to the power of 5 is 5 x 5 x 5 x 5, it gets tiring, right?

3, right, that would be 5 x 5 x 5 and now here folks, we can simplify between the factors that are in the numerator with the denominator here I don't have 3 factors so I'm going to simplify the bottom 3 with the top 3 that's left for me there are two factors five left and two factors five will be the same thing as 5 squared and then tell me something, wouldn't it be much easier to have done here o five of this exponent here o minus three of this one 5 take 3 gives 2 so Give what you're saying here in the property, we're going to keep the base and we're going, oops, subtract the exponents, I even made a line there, so I always subtract the exponents, it's that easy. So also see another observation that you can find exercises of this type in the form of a fraction but you can also find it like this in this horizontal writing indicating division by: and here how it will be 2 raised to minus 3 divided by 2 raised to 4 Go on, keep the base because the bases are repeating themselves and then I'm going to subtract the exponents, so it's going to be minus 3, then until you get a highlight here, - 3 - this minus here is the minus there that came from the property, right? - 4 I will put it in parentheses here just so as not to confuse you and now here is 4, right guys, it is But, right, you could do it straight away, right?

- 3 - 4, which I'm going to do, so keep the base and I'm going to add the exponents because here I have a debt of 3 and a debt of 4, - 3 - 4= - 7, right? Here in the next exercise the bases are the same so I'm going to keep the base and I'm going to subtract the exponents, which is the same case as there, right ? in the second exponent here, then minus look here where the danger lies, people, minus then this minus here is from the property so I opened parentheses, minus 5 to Gis but why here you don't because sign, because here when the number there's no sign, it's not because it's more, right, and if I did the sign rule, more with less would mean less Wow, I'm writing crookedly here, folks, it's difficult to write like that, it would be less, so that's why I didn't indicate it, I did it straight, right?

You 're already an expert at this part of the sign rule, right ? what is the result of the exponent 2 + 5, right because I did the sign rule it became + 2 + 5 so I will have the exponent 7 ready, simple as that and here when I have letters now we have an X same analysis scheme the base they are the same the bases are equal Ok you will keep the base and let's go minus 3 I will subtract the exponents, minus the property and then 5, same case as that there would be plus 5 do the sign rule and it will be minus and then you will have what X raised to how much? -3 -5 will be equal to - 8 so here are four cases, right when we are going to apply the property which is the division of power with the same base and now when we are going to talk about power of power which is our next property but not necessarily in that order, right people, so when you find a power of power, it's very simple what you're going to do, so analyze the example I'm going to show you now.

So here's one more time for you to check, how this property is applied, you know, the power of a power is very easy too, so let's do that whole procedure complete 5 to the power of 3, what does 5 to the power of 3 mean? Remembering that I'm going to start inside the parentheses, right, so it would be 5 x 5 x 5, right? 5 to the power of 3, now all of this here is to the power of 2 and now how do I do it I have to do this times itself so it will be what it will be 5 x 5 x 5 x 5 x 5 x 5 oh, right then see that here I have a factor, right, that would be it here and here I have another factor, ok and then how many factors five do I have here in total so putting them all together I will have 5 to the power of 6 one two three four five six factors, but no it was simpler to do it another way, you can visualize what you could have done here without having to do all this here, o Read the power definition of a power well, so what am I going to do, I'm going to keep the base and I'm going to multiply, o multiply the exponents are so if I did it directly, I multiplied 3 x 2, I would already arrive at 6 directly and I didn't need to go through this procedure here.

You saw what I do, that's people, so now you're going to do this directly for me, how much power will it give? and of a power, right, and there will always be one exponent after another like this, so here I'm going to keep the base and I'm going to multiply the exponents, so here I'm going to get minus 12 because 3 x 4 is 12 and minus this one is plus like this here is less minus with more it will give less, okay, but in this case here the base remains 2/5 and what am I going to do with the exponent again, right this one is the easiest of all just multiply the exponents so here I I'm going to get minus 10 and here people see that I 'm going to keep this fraction in parentheses because I keep it in parentheses because this exponent is both an exponent of the number 2 which is the numerator and the exponent of the denominator so that's why I put parentheses now if I had written it like this, minus 2/ 5 to the power of minus 10 like this here like this here, it's indicating, it's indicating that this minus ten is the exponent of just this two and it's not the exponent of five. So this writing is meant to represent that the minus ten is the exponent of the entire base, it will be incorrect, so be very careful, then we have to put parentheses Okay, so mark it well, look at this writing here, so this here is different from this here, another observation for you, and in this last one here, there is an exponent exponent exponent when there are three, how do you normally keep the base which is x and then you Go Multiply the exponent just multiply and multiply 3 x 2 by 6, 6 x 5 will give 30 and then I do the rule of signs this three is more but with less than less and this other five is more minus with more will give less then it will be equal to minus 30, okay guys, an observation that I need to make with you so you don't get confused.

Look, here is an extra example for you when I have 3 to the power of 2 to the power of 4 an extra example here how would you do this case here you would keep the base and you would multiply the exponents would result in 8. Okay? Now, pay close attention, if you had an exercise like this, 3 to the power of 2 to the power of 4, would you be able to observe any difference in this writing in this writing ?

Gis, but this doesn't have the same answer, because here I apply the power property, which is the power of a power, just multiplying them exposes here what you have here, you can see the exponent here, what is the exponent Guys, what make up the exponent, the exponent is composed of a power that is 2 to the power of 4, you see here the 2 to the power of four, so for me to solve this type of question, first I have to calculate 2 to the power of 4, how much will 2 to the power of 4 give? 2 to the power of 4 will be equal to 16. Because it is 2 x 2 x 2 x 2, then it results in 16, so, look closely at the two results, see how the results are different, people, so you have to be very careful, yes, when I have here with the parentheses and when I have a power that has the exponent here, being another power, mark it very prominently, very big, right, so you don't forget and don't make mistakes in your activities, agreed then?

Then the next example of the properties that we are going to see, it talks about the distributive property in relation to multiplication and division, it is also very simple. So see that I said the distributive, right in relation to multiplication and division but here I wrote power of a product because I wrote it differently, you know because when I have the power of a product I will make a distribution there with the exponents so That's why I said distributive. So this first example, I'm going to do the classic one, right?

Let's show where it came from. So see that here I have 7 to the power of 2 and this 7 to the power of 2 also has an exponent of 2, so I will have two factors of it, so it will be 7 to the power of 2 times 7 to the power of 2, okay, let's do the same thing with 5, which will be 5 to the power of 3 times 5 to the power of 3, two factors because of the exponent now I can apply that property of powers when I have multiplication of powers with the same base what does it do, let's see if you paid attention at the beginning of the class very well, we will then keep the base that is repeated and add the exponents 2 plus 2= 4 times, I will do the same here with the base 5 now keep the base Do you remember that I said that I could only apply this property when the bases were equal so I'm applying these two are equal and now these two are equal, okay, I couldn't do everything at once because they weren't all the same, right, so here it will be 5 to the power of 6 because three plus three So see that here I have the final result so in these types of exercises the results will be in the form of two powers of a product of two powers. Okay, you don't need to develop it because the exercise will tell you to stop there, so what can you conclude, you said you could have done to be faster, have carried out a distribution, right, a little shower, do the shower, it's true, little shower, we do a little shower here right, with the exponents, 2 times 2 which will be 4, 2 x 3 which will be 6, that's it, look how much easier it is than doing all this here but, you know, I like to show it.

So let's go in this case here what will it look like five times 2 So the bases I have already copied now let's do the distribution of the exponents I will do 5 x 3 which will be equal to 15 and 5 x 4 which will be equal to 20 and I have the final result here so remembering that I do the multiplication, because it's a distribution, right, so do the multiplication here now with a negative number I just want to see heh, so it will be six times three and let's work on the exponents - 2 x - 2 - minus gives more so we will have exponent 4, - 2 x plus 5 will be equal to minus 10 ready closed result and here in the last one I now put the letter right because it has to be done several cases for you to identify minus one Ah, I have the bases, I forgot to write, minus one times five will be minus five and minus 1 x 2 will be minus two, oh guys, so here I applied the distributive in relation to multiplication, right? Here see that between these powers we have a multiplication and now we are going to do the examples when we have the division, shall we? Now in the distributive in relation to division then we have the power of a quotient because everyone already knows that it is related to division.

So see that I have a division in the form of a fraction which is represented in the form of a fraction and these others cases I wrote differently so to give you an idea of ​​all the cases, what happens here, let's do that long procedure 7 to the power of 2 to the power of 3, right, so I'm going to do what I'm going to do, which is 7 to the power of 2 times 7 to the power of 2 times 7 to the power of 2, 3 factors, right, and the same thing in the denominator 5 to the power of 3, so it will be 5 to the power of 3 three factors because look there the exponent times 5 to the power of 3 times 5 to the power of 3 now what What I can do now I can apply that property multiplication of powers of the same base keeps the base and adds the exponents. So it will give two plus two plus two which will be equal to 6 I don't even know why I made it so big and here again it keeps the base and adds the exponents so it will result in 9 and now I can keep the base and subtract the exponents? No, right, because the bases are not the same, so here I have my final answer and can you already identify what I could have done, straight away without having to do all of this?

Oh, do the distributive too, right ? It's both the numerator and the denominator, so there's the importance, right? And I multiply it three times three and I'm going to get 9, so quick, so let's take advantage of the way, then we're going to do minus 2 times 3, it's going to be equal to 2 to the power of minus six.

because minus 2 times 3 will be 6 - with more or less there divided because here I have a division and here I also apply the rule So it will be 5 - 2 x 4 will be equal to minus 12 guys I'm done you can leave it written So, well, as you can also write here in the form of a fraction, in this other case again, I will apply the distributive minus 2X minus one, it will be positive two because minus with minus gives more and here I will have then divided by minus 2 times 3 will be minus 6 and here folks, can you see anything that I could have done, what I can do, it's still very good for you who observed that here I have a division of powers with the same base, so as soon as I can do it here, I'm going to keep the base and I'm going to subtract the exponents I'm going to do here so 2 - minus 6 this minus here is from the property and it's good not to mix it because then you'll have this perception of which property to use in each case less with less will give more so I'm going to do two plus six and two plus six will result in 8. Gis but if I wanted to have applied it here at the beginning I could have applied this and when I have the division I could keep the base and subtract the exponents you could have done it here and then Having applied the second exponent from the outside, of course you could, but all cases will reach the same path, that's fine. All paths will reach the same result, I said the opposite and in this last case here, base two, what am I going to do, let's distribute the exponents - 3 x 5 is negative 15 and here I'm going to do negative three times negative two so it will be three to the power of plus six because less with less is more.

So see that here you have another property and here it's already mixed with people so it stays there, you know, marking these tips because it goes on, but at some point in the exercises you'll have to say: and now what do I have to do but you have everything is marked so you're going to rock it, there won't be any problem, right, and once again asking you to take the opportunity to subscribe to the channel if you're not already subscribed, then you'll receive notifications, right, activate the bell to receive notifications of classes that the Gis publish weekly and then I managed to do all the activities and also very important, give a thumbs up and be sure to watch the other classes that I have already explained about powers, the definition, scientific notation, which is also very important, powers with a negative exponent and all these classes you will find the link here in the description, right guys and also if you are looking for content on the Gis channel, just type the name of the content and add the name Gis in front or giscomgiz which goes straight to that class that contains the content you you need to study and also share this video with your colleagues so everyone knows the properties of powers and I'll see you in the next class, bye. . .