Hi guys, do you know what an exponential equation is and how to solve it? If you don't know and want to learn or remember, I invite you to watch this class and I'll explain all the procedures, let's go? So welcome to my channel, I'm Gis and before I explain how to do an exponential equation I want to ask you two things, subscribe to my channel so you can receive directions to the classes I publish and leave a thumbs up for me, okay guys?
agreed, right? Now I'm going to explain to you what exponential equations are, it's all that equation that has the unknown as an exponent, look here at x, it's there in the exponent, now it's different from when I studied an equation from the first degree in which I had, for example, 2 x plus 4 for example = 8 what I wrote here is considered a first degree equation, look at the unknown, where is the one I'm doing now that you're going to learn is an exponential equation, because the unknown that is x, look where it is, it is in the exponent, you attended the class in which I explained about potentiation, there are all the properties, I explained how to make a power of a natural number, a power with a negative exponent, I explained all the procedures, I will leave the indication to you, so that Can you remember these concepts ok? so I determined that this is an exponential equation look here also x is in the exponent so it is also an exponential equation, it is not right the equation is considered exponential because x is the exponent now think like that how do I solve it so the procedure is easy to solve, you see that here there is a base 2 and 2 is a prime number, and you see that here it is 32 and the base here the exponent here would be 1, the base is 32, the exponent would be 1, and then we like Should I proceed now we are going to do the decomposition of the number 32 the decomposition of the number 32 into prime factors you don't remember how to do this decomposition so I will also leave the indication of the class so that you can remember these concepts so I will do the what is the prime number that divides 32 by two 32 / 2 is 16 sixteen ok, 16 can be divided by 2 which is 8, 8 can also be divided by 2 which is 4 which is divide by 2 , which is 2, which is possible to divide by 2, which is 1.
Well, look guys, when I did the decomposition of the number 32 here using the prime factors, it turned out to be 2 x 2 x 2 x 2 x 2, so here I can represent it as 2 to the power of 5 because it is 2 x 2 x 2 which results in 32, right, so I 'm going to write it like this here now 2 to the power of x equals now instead of writing 32 I'm going to write its power form here which was 2 to the power of 5 is now here I managed to leave the two bases, base is the number at the bottom, exponent is the number at the top, so now that I managed to leave the two powers with the same base I can ignore the bases, I I can say goodbye to them, okay, I can remove the bases and I will make the equality between the exponent so here I will have that x will be worth five so I managed to solve my exponential equation the value of x here is 5. but it was very easy to I could have done it already mentally, it's not because what number I'm going to raise to base two which gives 32, then you can think 2 times 2 is 4, 4 x 2= 8, 8 x 2= 16, 16 x 2= 32 That would be five, so I could have done it straight away, but I'm explaining the procedures to you so you can already apply these here when the equations got a little more complicated, right guys? Look at this one here 3 to the power of 2x = 81 are here we know that we can't change this number three, we can't decompose it because it's already prime, it's already prime and 81 can I leave it written in the form of a power with a base of three does not necessarily need to be prime, but I always have to keep an eye on the base I have here, which will be a base 3, so look, let's factor 81 people, factor, do the decomposition into prime factors, so it will give Here, for three, it will be 27, Gis wants to advance the work, for three, it will be nine, for three, it will be three, for three, it will be 1, okay, I'm so practiced at doing decompositions that I can do it very quickly.
So here when I did the decomposition of 81, I wrote 81 in the form of power is the same thing as writing it as 3 to the power of 4, right, so here look what I 'm going to do now it will be 3 to the power of 2 x equal, instead of writing 81 I'm going to write three to the power of four so see that the The objective was achieved, which was to leave the two bases the same, just like I did here in the two, I left the two bases the same so I could ignore them, so I could say goodbye to them, okay, ignore them in a good way, leave them behind so I wouldn't need to use them anymore . count, okay guys, so here I can take 3 and take 3, right, bye for them, what's left? The two exponents are left, so I make the equality between the two exponents 2x = 4 now here guys I have a first degree equation that would be equal to this one here, x is not an exponent but it is the normal number it is the number that is being multiplied by two, okay, so you were able to observe the difference between when x is an exponent and when x is a number that is being multiplied by another here, okay, so how do I finish this calculation?
Now it's easy, if it's multiplying, I divide by 2, 2 divided by 2 will be 1, so 1x and 4 divided by 2 will be 2, so that means that in this question, x will be worth 2, right? Let's do other examples, shall we? Guys, let's solve the next examples that I brought, look here, now I brought a fraction so we can do factorization, decomposing each number is easy, when I have a fraction and I look here, here it is 8, here it is 2, here it is 27, here it is 3, I think well I have to do the decomposition of the number 8 which is the numerator and the decomposition of 27 which is our denominator so we are going to do this separately then factoring the number 8 doing the decomposition into prime factors ok it will be 2 which will be 4 by 2 is 2, 2 is 1 so 2 to the power of 3, right then I 'm going to do the same with 27, 27 is 3, 3 is 9, 3 is 3, 3 is 1 so here is 3 to 3 OK, so you can see that the exponent of the two numbers is the same, the exponent of the two numbers is 3, so I'm going to write it like this, it will stay, this one stays the same, I won't change it, but here now I'm going to write it as two instead of writing 2 to the power of 3 and 3 to the power of 3 instead of writing each one with its exponent as an exponent, it's the same as the properties of the powers, I can write an exponent just for both of them, but then I need to put the parentheses right and then What do you conclude from this base 2/3 base 2/3 are equal so I can take the base here what is left there is equality between the exponents so what is left is that x = 3 ready guys look how quick it is to solve this here, let's do it the next one now, see 16 here and 1 here and now if I factor 16 people, look here, if I factor 16 it gives 2, right, it gives 4 by 2, which gives four, look, I'm skipping a step guys, no one told me which gives 8 by 2 which gives 4 by 2 from 2 by 2 gives 1, so this here is 2 to the power of 4 but and now it gives 2 to the power of 4 here and this side is not going to be enough for me to be able to remove the base what am I going to have to do, so that means the strategy I'm going to develop and this isn't it, you attended the class in which I explained the properties of powers, all those cases, I'll leave it for you in the description, the link is ok, you can access this class and remembering these concepts people is very important because you see, 16 to the power of x minus 3 if I factor 16 I arrive at 2 to the power of 4 which is a base 2 and on this side there is a base 1 wow so that means the following I have it so that every number raised to zero, any number that I put raised to zero, as long as this base is not zero, okay, that's all I explained there in that class and I'll leave the indication for you in the description here too, so every number that I I raise this number to zero and it results in one, so it means that if I write here instead of writing the number 1 I write my 16 which I have my eye on here, right, it's one eye on the cat and the other on the fish, that's not how you say it the saying has to keep an eye on both things so if you write this here as being 16 to the power of 0, 16 to the power of 0 is how much 16 to the power of 0 returns to 1 then it means that I am using a power with exponent zero to result in 1 so I'm not changing the value of anything I'm just changing the way I write it, for example I can write the number 6 as being the product of two by 3, 2 x 3= 6 okay So if I'm doing another writing but the result is the same it doesn't change it will always be 1 every number raised to zero is 1, and on this side what I do I'm going to leave it as 16 even x minus 3 a but There's no need to factor, just leave the prime number here, don't remember what I said at that time, it doesn't necessarily need to be the prime since I've already managed to leave the two bases equal bye, bye base that what's left is left that x - 3 = 0 and now how When I finish this calculation, x is less, so I'm going to add three more, it will be x = 3, right, I throw it over there, we talk like this, right, the most practical way, so I can go again, look what a coincidence, x was three here too OK, folks, so you saw how it's possible, yes, that's why you have to know the properties of powers, all those concepts, right?
Shall we do more examples? Guys, now look at this example I brought, there is a fraction and on this side there is no fraction and here it is base four and here it is base 32, can I write 32 using a base 4, think about it, 4 x 4 gives 16, then 16 x 4 = 64 ran away from 32 so it's not possible, so in this case, as I can't leave one base here the same as another, I have to move both bases, so it's easy, isn't it, look here, it's four, look here I wonder if 32 will happen, let's decompose it , so why, right? it will be 4 by 2 it will be 2 by 2 it will be 1 so this 32 here is the same thing as 2 to the fifth power ok so I'm going to write it here now, 32 will be 1/2 to the power of 5 ok, ok and now guys what what I can do if I also perform the decomposition of 4, it's here, 4, taking advantage of the same here, 4 is what is 2 raised to 2, right, because 2 x 2 is 4, so it will be 2 raised to 2 and raised to x that was there, to x that was there right and now you remember the properties of powers that I have power of a power so I can multiply so here it will be 2 raised to 2x =, here I could have done it straight away, okay?
good people, just showing you the procedure and here people now the base is 2 and the base is 1/2 oh but there is that case of negative exponent remember I do the inversion remember I explained I said it here it makes the whistle I had to make the whistle for real so what does it mean to make the whistle, the 2 will go up and the 1 will go down I invert the base and from the moment I invert the base I change the sign of the exponent is the opposite the exponent is no longer five so when I did this inversion it will be minus five but guys this 1 here will not change the value of my power if it stays there or not because 2 divided by 1 is 2 so look I'm going to take it out of here, look how beautiful it is now, base 2 with base 2, so bye bye to the base and I do the equality between the exponents, so it will be 2 x, and then I end up in a first degree equation that is being multiplied, so if the two are being multiplied by x, I will divide by 2 and here I also divide by 2 so here 2 / 2 will give 1 x equals minus 5 / 2 or you put it in decimal form or leave it as a fraction then it will be -5, let's go writing it correctly here because it makes me uncomfortable to write it like this, now it looks beautiful, so here in this equation the value of x is -5/2, right people, look here, it's simple, so always applying the decomposition and I always need to remember those little rules In addition to the powers, this case here also has a root here and then how is it the case of writing the number that was in the root without being the root using the fractional exponent, you don't remember so I'll leave the indication so you can Go back to this class too and remember these concepts, okay, so here 49 let's factor 49 or can I leave 343 in a base 49, let's start with 343, 343, do you know folks, the practical way to find out what number gives divide here that 343 is divisible by what number I put here that gives the exact division so we have the divisibility criteria there in case you want to remember because it is a more practical way you don't need to keep doing the division I think it works with the calculation will be exact or not, if it's exact, I'll do it, if it's not, why am I going to waste time doing it and seeing that it won't be exact, okay, so I'll leave the indication here of the divisibility criteria class to you. So, this number 343 is not the number divisible by 2 because it is not even, is it the number divisible by 3? 3 plus 4 gives 7 and plus 3 gives 10, it's not because 10 isn't in the three times table, is it the number divisible by 4, why not by four, isn't it because here I only put a prime number, right, by 5 it's worse by 5, no It's because here, to be divisible by 5, it has to end in 0 or 5, it's not 6, it's prime 7.
Is it possible for 7 ? 7 but you're asking what the divisibility rule is . my class doesn't have the number 7 rule, okay because the books also don't include the number 7 rule, okay, so by dividing 343 by 7 I'll find 49, here 49 there and then I'll continue, right, I won't stop halfway gives seven and seven gives 1, so 343 is the same thing as 7 to the power of 3 and you see 49, what is 49 now thinking, wow, in base 7 49 in base 7 is look here Oh, 7 to the power of 2, right, so here I'm going to write 7 to the power of 2, which is 49, and this here will be raised to the x + 1, I can't forget this exponent here, ok, the same, now it will be the fourth root, 343, I saw that it is what is 7 to the power of 3 and now what do I do, because this 7 is inside the root and this one is not there, here comes the rule of the fractional exponent here it will be so seven here I can now apply it so a distributive 2 times _ from the root there it will be seven outwards and then the exponent will be a fraction so it does this: whoever is in the sun, thinks that this four is in the sun, whoever is in the sun goes to the shade, whoever is in the shade comes to the sun.
so writing the fourth root of 7 raised to the cube is the same thing as me writing the power with a fractional exponent 7 raised to 3/4 and then look how beautiful 7 and 7 bye bye for them are left so 2x plus 2 = 3/4 and then What do I have to do now, this 2, which is more, I'm going to subtract it, it will be 2x = 3/4 - 2, I put 1 here so I'm going to put 1 here to do the mmc process, which will be the mmc between one and four, what is the mmc between 1 and 4 people? The mmc between one and four will be 4 ok, let's write here 4, 4 for each of them and then from 1 to 4 I multiplied by 4 and from 2 to 4 of 2 here I also multiply by 4 node, from 1 to turn 4 I did times 4 so 2 times four will be 8x from four to four it's the same so I just copy 3 of 1 to get to four I did times four so here it is also times four which will be eight and then I remove all the denominators because as it is an equation that I do on one side I also do it on the other and it doesn't change so what remains is that 8x = 3 - 8, 3 - 8 is minus 5 and now to finish everything here I divide by 8 because this eight is multiplying I cut the eight here so it means that the value of x, the value of x in this question will be what, the value of x will be minus 5/8 ok guys, look this one takes a little bit of work for us to do the x so it's worth 5/8 but I wrote it here a little crookedly, below it's ok so how do you do the exponential equation so I make the two bases the same so I can say goodbye to them and then I make the exponents equal, ok Guys, oh and I'm going to invite you to watch the next class in relation to this content because we have cases that are a little more specific and a little more difficult that I'll explain in the next class, okay, so be sure to watch the next class so that you you can know everything about exponential equations and leave a thumbs up for Gis, thumbs up for Gis and subscribe to my channel if you liked the class and then you can receive notifications of the next classes I publish. Until next class guys, bye.
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