Monomials Multiplication & Division

Hello everyone, welcome to our Mat-E-Sip YouTube channel where we make math easier to understand with our free Tagalog tutorials. Our topic for today is all about multiplication and division of monomials. But before we start our lesson, if you're new here, subscribe so you'll be updated on our lessons. And to my fellow teachers, we also offer our editable PowerPoint presentations to save you time in your lesson preparation. You can check the link below or message us on our Facebook page. Alright, so let's get started with our lesson. Okay, so class, at the end of this lesson, you will be able to multiply and divide simple monomials leading to the derivation of the loss of exponent. Okay, so let's proceed to multiplying monomials. What is our rule when it comes to multiplication? Okay, so let's have this exploration first. We will observe this example so that at the end, we are able to derive what is the rule when it comes to multiplication of monomials. So as you can see, we have here two monomials multiplied to each other. We have here x raised to 3 or x cubed times x raised to 4 or to the power of 4. So what will be its answer? The first thing we are going to do is to expand. So the way to expand this is, since this is x raised to 3, we are going to write down 3x. Since x raised to 3 means multiply x by itself 3 times. That's why we have here 3x. Paano naman po yung x raised to 4? Ilang x po yan? Correct, there are 4x. Okay? Ito na po yung tinatawag nating mga factors. Ano po ang factors? Factors are numbers or variables multiplied together to form a product. So what are we going to do with these factors? We are going to combine. Combine the factors. So in our first monomial, x raised to 3, we have 3 factors of x. And then x raised to 4, we have 4 factors of 4. We are just going to add them. We are going to combine them all. So how many x are there? There are 1, 2, 3, 4, 5, 6, and... So, ano po kayang magiging sagot? Yes, it's actually x raised to 7. The question is, do the expressions have the same base? Pare-parehas po ba yung base nung dalawang monomial? Yes, ano po yung base nila? It's x. Another question, what happened to the exponents? Ano pong nangyari sa ating mga exponents, 3 and 4, bakit siya naging 7? Okay, correct. Ang ginawa nang natin dito is we... add the two exponents. That's why our exponent on our answer is 7. Okay, another example here, we have here x raised to 4 times y raised to 6. What will be our answer? So if we're going to do the same process earlier, we are going to expand. So we have here x raised to 4. How many factors po yung ating x dyan. There are four factors of x. And then next one, Ilang factors po yung ating y raised to 6? Okay, ganun din po. Anim, no? Depende sa ating exponent. So, that is the expansion. These are the factors. Now, we are going to combine these factors. So, how do we combine these two factors? Okay, the answer here is x raised to 4, y raised to 6. May nagbago po ba sa kanila? Okay, parang pinagdikit lang po natin sila, no? We cannot combine these two. Because they have different base. Okay? Magkaiba po yung base nila. So we cannot just make this 10, no? Hindi natin pwedeng i-add yung 4 and then 6 to make it 10 because they have different base. So again, do the expressions have the same base? Hindi po. Okay? What else? What happened to the exponents? As is po yung ating mga exponents. Wala pong nangyari. We just write them. beside each other. Wala na po yung multiplication sign. Yan na po yung mismong sagot. Okay, so basing on those two explorations, we can now derive the rule on multiplying monomials. Okay, so ano po yung rule natin? Eto po. So we have here x raised to m times x raised to n is equal to x raised to m plus n. Meaning, if we have here a base which is x and then we have an exponent which is M And then we will multiply it with another monomial, which is x raised to n. Ang gagawin lang po natin sa mga exponents nila ay i-add lang po natin. So whether magkaiba or parehas ang kanilang exponents, we can just add the exponent. Ang tawag po natin sa rule na ito ay, product rule. Okay? And take note, this rule only works if the bases are the same. For example, we have x or a or any number. As long as the base is the same, we can apply this rule. If their bases are different, just like in exploration number 2, we cannot just add the exponents. Kokopyahin lang po natin. Okay? So, ito po ang product rule. Okay, to give you an example, we have here an activity. Let's find the product of the following monomials. Okay, again, so you can always pause this video if you want to answer this on your own first. But if you want to watch first these examples, you just follow along while we answer each number. Okay, so first, number one, we have here x raised to 8 times x raised to 2. So following our product rule, let's check first kung parehas po ba ang kanilang base. Alright? Do they have the same base? Okay, tama. Now, we will just add the exponent. So, therefore, our answer is x raised to 10. Okay? 8 plus 2 is equal to 10. And then, kinapi lang natin yung base nya. Next one, do they share a common base? Yes. And then, we just add 2 and 3. So, the answer is a raised to 5. Next, parehas ba ang base? Yes. Okay, not necessarily a variable. So we'll... 2 is our base and then our exponent is 2 and 5. So we just add this, we will get 7. You can further simplify this by multiplying 2 7 times to itself so that you can get the final answer. But this can also depend on your teachers. Next one, we have here y raised to 4 times y. So since this exponent is not there, it means it is raised to 1. Therefore, you just add 1 and then 4, so magiging y raised to 5. Again, kaya naging 5 kasi may 1 siya dito. Next, we have here 3 monomials multiplied. So, paano ito? So, do they share a common base? Yes, they share x and then y. Dito x and y, x and y din. So, first, let's have x. Ano ba ang exponent ng x? Exponent nya dito is 1. Exponent nya dito sa second is 2. And then exponent nya dito is 1. So you just add those 3. 1 plus 2 plus 1, it will give you 4. Next one, we have y. Exponent nya ay 2. Dito naman ay 1. Dito naman ay 1. So, ang sagot natin, ganito. Alright. Ia-add lang natin lahat ng exponent. Therefore, our final answer is x raised to 4, y raised to 4. Okay, let's have more example. We have here number 6. We have here monomial 9x raised to 3 or 9x cubed y raised to 4. And then, multiply it to 4x squared y cubed. So, ano pong pinagkaiba nito sa mga examples natin kanina? So, if you observe, meron na po tayong mga numerical coefficients, yung 9 and then 4. So, what do we do with this? So, just like in our exploration, we can expand this just to demonstrate. Para lang ma-explain ko sa inyo, if we expand all the factors, we still have 9 and 4 as factors dito. So, mumultiply lang natin sila. Okay? Yung mga factors ng x cubed, nandito, 1, 2, 3. Factors ng x squared, 1 and 2. Ganun din po sa y. Alright? So, all you have to do is to multiply 9 and 4, you will get 36. And then you combine all these factors, x raised to 5, y raised to 7. 3 plus 2 ito, kaya naging 5, and then 4 plus 3 to make it 7. So again, just multiply the numerical coefficients if there are coefficients. Okay, kailangan pa bang isulat natin itong expanded form? No, no need po, no? Sinulat ko lang po yan for demonstration. You can go away with this and directly write the final answer. Next, number 7 we have here. We still have numerical coefficients. Negative 5. x cubed y squared times 3, x cubed y raised to 4. So let's multiply negative 5 by 3, we will get negative 15. And then x cubed times x cubed, just add 3 plus 3, we get x raised to 6. y squared times y raised to 4, 2 plus 4 is 6. Okay, negative 15x raised to 6, y raised to 6. Okay, for our last example, we have here negative... 6 a raise to 3 b c raised to 4 times negative 3 a b c d. Okay? So, what's the difference between these 8? So, if you observe, here in the second monomial, we already have an extra variable that's not here in the first monomial. So, what do we do with this one? Just like that, all we have to do is to multiply their common elements, their variables. or even the numerical coefficient. So negative 6 times negative 3, we get positive 18. a raised to 3 times a is a raised to 4. b times b, since their exponent is 1, we just add 1 plus 1, you get 2. Okay, b raised to 2. Next, c raised to 4 times c raised to 1. 4 plus 1 is 5. So we have c raised to 5. And then since D has no partner here in the first monomial, you just add or you just copy our D here. Okay? Therefore, our answer is 18, a raised to 4, b squared, c raised to 5, d. Not all bases are the same. It doesn't need to be the same for all bases here. Okay? So those are our examples for multiplying monomials. Okay, so let's now proceed to dividing monomials. But before that, let's have first a short commercial break for our dear teachers. Okay, so first let's explore these examples so we can also derive the rules in division of monomials. Okay, so we have here the first fraction. As you can see, we have 5 over 5 or 5 divided by 5. So, what is the answer here? Okay, yes, correct. It is 1. Okay, in short, we can just cancel this out. Okay, meaning it will become 1. Okay, it is equal to 1. And also with this one, negative 3 over negative 3, we just remove or cancel this out. Magiging 1. Yun po yung ginagawa natin when the numerator and denominator are the same. So, nagiging 1 lang po siya. Hindi po, 0 ha. 1 lang. Okay? Take note of that. So, next, what can you say about its dividend and divisor? So, where is the dividend? The one in the numerators. So we have 5, negative 3, and 10. These are all the dividends. And then the divisor are the one in the denominator. So it's the same. We have the same numerator and denominator. So we have the same dividend and divisor. Next, what have you noticed in the quotients of the given integer? So what is their answer? All of them are 1. Alright, so... Let's explore more on this example. We have here a variable x with an exponent of 5. And then sa denominator, a variable x with an exponent of 3. So how do we divide this? We will expand first. Kunin natin yung mga factors. Okay, so we have here x. Ilan po bang x ang factor nito? So there are 1, 2, 3, 4, 5. That's why it was x raised to 5. Next one, how many x dito sa denominator? Yes, 3. Okay? Now, just like in dividing those fractions, we can actually cancel out kung ano yung nasa numerator na nasa denominator din. So, we can cancel this first pair, second pair, and then third pair. So, ano na po yung naiwan? We have 2x on the numerator. So, yun na po yung ating sagot. Since dito sa baba, ano na po ang naiwan dito sa denominator? Okay, hindi po siya zero. Okay, may naiwan pa rin po dito na one. Hindi lang po naisulat. Okay, therefore, itong dalawang x na to ang denominator nila ay one. So just combine these factors, magiging x raised to two na po siya. Okay, so what did we observe from this? Do the expressions have the same base? Yes, parehas po ang base. Next, what happened to... to the exponents. So, anong nangyari sa mga exponents natin? Yung 5 tsaka 3 dito, ang naging result sa final answer ay 2. So, what did we do with 5 and 3 to make it 2? Okay, correct. We just subtracted those two exponents. Okay? So, earlier in multiplication, we added. Now, we will subtract. This is what we call the quotient rule. In product rule, we added exponents. Here in quotient rule, we will subtract. It's the opposite. So if we have here x raised to m or any exponents divided by x raised to n or any exponent, we just subtract m and n. So it will be x raised to m minus n. By the way, this rule only works if the bases are the same. Parehas din po sa product rule. Dapat parehas po ang base nila. Kung magkaiba po ang base nila, you cannot apply this rule. Okay, another thing to take note is ito po. Pag negative yung exponent, halimbawa, We have here x raised to m and then x raised to n. Pag pinagsubtract mo sila, ang lalabas ay negative. Anong gagawin natin? Okay, yung magiging numerator natin is 1. Tapos yung variable na may negative exponent mapupunta dito sa denominator. At ang kanyang exponent ay magiging positive. So, mamaya magbibigay po tayo ng example nito. Alright? Where m is less than n. Okay, so we have here these examples. So again, if you want to answer this first, para lang mataray mo, matest mo, you can do so. Pero if you want to watch first these examples, panoorin nyo po muna yung gagawin natin. So number 1, we have here m raised to 5, n raised to 7, o raised to 3 over m raised to 4, n raised to 3, o raised to 3. So the first question is, pare-parehas ba ang kanilang base? Yes, pare-parehas. Now, can we apply the quotient rule? Yes. So we have here the answer, m raised to n, I mean, mn raised to 4. Okay? Paano po nangyari? 5 minus 4 is 1. That's why we have here m raised to 1. Do we need to write 1 here? No. Next one, 7 minus 3. is 4. So we have here n raised to 4. Next, dito naman po sa o, 3 minus 3 is 0. Okay? So any number or variable with an exponent of 0 is equal to 1. Hindi po 0 ha, 1 lagi. So do we need to write 1 here? No need. Okay? Kaya ito, cancel out na po yan. Pag parehas po ang exponent ng nasa numerator and denominator with the same base. You just cancel out. Okay, dito sa number 2, okay, nabigay ko na po yung sagot. We have here 8x raised to 5 over 2x raised to 2. So, pag may mga numerical coefficient, tayo dito, we just divide. So, 8 divided by 2 is 4. And then for our x, 5 minus 2 is 3. That's why 4x raised to 3. Okay, next number 3. Ano yung magiging sagot natin? So, we have here negative 3. d f raised to 2. So paano nangyari yun? Negative 21 divided by 7 is negative 3. Next one, d raised to 3 over d raised to 2. 3 minus 2 is 1. So you just write d. Okay, raised to 1 yan pero huwag na nating isulat yung 1. Next one, e raised to 7 and then meron ding e raised to 7 dito sa baba. You can just cancel this out. Okay, cancel this out. Next, f raised to 5. over f raised to 3, 5 minus 3 is 2. That's why f raised to 2. Okay, number 4, we have here negative 54 over 9. Okay, pag dinivide natin ito, magiging positive 6. Okay, that's why we have here 6. Dito naman sa x natin, we have x raised to 5 over x raised to 3, 5 minus 3 is 2. Okay, and then dito, y raised to 8 over y raised to 7, 8 minus 7. is 1. That's why we have y raised to 1. We don't need to write 1. Okay? Ganun lang po siya. Okay, let's have number 5. This time, ano po ang pinagkaiba nito sa previous examples natin? If you notice, yung nasa numerator natin, tatlo po yung variable diyan, while here in the denominator, dalawa lang. So, meron pong z na walang pares dito sa denominator. Okay, so if we divide 36 by negative 4, our answer is negative 9. Okay, next one. We have here x raised to 3. raised to 2 here in the denominator, 3 minus 2 is 1. That's why we have here x only. Next, y raised to 3 over y raised to 3. Cancel this out. So, y is gone. And then we just copy z here in the numerator. Okay? So, that's it. Next, for number 6. Okay? Let's find out what we're going to do here. In these examples, from 6 to 8, you have to pay attention because this is different from our previous examples. If you notice, the ones in the numerator have a smaller exponent as compared to the ones in the denominator. So eventually, our answer will be negative. Remember, 2 minus 5 is negative 3. And then 3 minus 4 is negative 1. So our answer is this. x raised to negative 3. y raised to negative 1. Pero meron po ba tayong polynomial na negative ang exponents? If you go back to our previous lesson on what polynomial is? Okay, wala po. So ang gagawin natin, i-co-convert natin ito sa polynomial. So using our quotient rule, yung nabanggit po natin kanina, na negative exponents, ang gagawin lang natin is ibababa natin ang mga ito. So, magiging fraction na po siya. Okay, and this fraction, if you notice, meron na po tayong numerator na 1 sa taas. Tapos, itong mga ito na negative ang kanilang exponents, ibinaba lang po natin sa denominator. Now, pag binaba na natin sa denominator, yung negative na exponents magiging positive na po siya. So, we have here x raised to 3, positive 3, and then y raised to positive 1. Okay, instead of... negative 1. Okay? Ganun lang po siya. Now for number 7, we have here another example kung saan may mga variables din tayo na mas maliit ang exponents as compared dito sa baba. Take note, negative po yung ating numerator and then positive yung ating denominator. So pag ganito po, laging ang sagot natin is negative. Okay? Negative divided by positive is negative. So the next one is we will now subtract the exponents. So this will be its answer. Negative 3a raised to negative 1, b raised to negative 2, and then c raised to 5. Since there's no c here in the denominator, we just copy the c raised to 5 here. Next, since we cannot have a negative exponent, we will lower the variables. So, hindi po lahat ibababa. Yung mga may negative na exponents lang po ang ibababa natin sa denominator. And then, yung numerator nya magiging 1. Pero, in this case, we have a negative 3 numerical coefficient. So, yun na po yung magiging numerator nya. Okay? Yun na po yung magiging numerator nya. So, we have negative 3. c raised to 5, kasi ito positive naman po yung kanyang exponent, no need to bring it down, over a times b raised to 2. Okay, since itong dalawang variables na ito, sila lang po yung may negative na exponents. Okay, for number 8, we have here negative 9 over np raised to 4. Paano po nangyari yan? We don't need to write this part, yung mga... May negative na exponent, pwede natin idiretso na sa final answer. Kung saan yung mga negative exponents natin is ilalagay na natin dito sa denominator. Okay? So again, 72 divided by negative 8 is negative 9. Okay? Yun po ang nasa numerator natin, yung numerical coefficient. Next, yung mga variables po natin, we have here m raised to 2. m raised to 2 din po dito sa denominator. You can just cancel this out. So, huwag na po natin idagdag sa sagot. Next, n raised to 2 over n raised to 3. Since ang sagot natin dito is negative 1, ang kanyang variable ay nasa denominator. Okay? Next, p raised to 3 over p raised to 7. Magiging negative 4 din po yung exponent dito. So, ilalagay din po natin siya sa denominator. Okay? That's why. So, ito po lahat yung mga examples natin. I hope you understood. You can backword if there are numbers or items that you haven't understood yet. So after this, I'm now going to show you exercises so you can also practice on your own. Okay, so this is now your activity. You find the product or quotient of the following monomials. So part A is for the product and then part B is for the quotient. Okay, I'm going to give you a few minutes or take your time as much as you need. You can pause this video so you can do it on a separate sheet of paper. And then you can just come back later to check your answers. Okay, good luck and God bless your answer. Okay, I hope you are now ready. I'm now going to show you the answers for this activity. Okay, so for letter A and letter B, ito na po yung mga sagot. Okay, pakicheck na po ang inyong mga answers and then I hope nakuha nyo po yung mga tamang sagot. Okay, so again, for our next lesson, we will now have multiplication and division of binomials and multinomials naman. Okay, so I hope to see you again on the next one. Thank you for listening. God bless and bye-bye for now. Kung meron kang natutunan sa video na ito, please click the subscribe button to support this channel. Para mas matulungan ulit kita sa mga susunod mo pang math lessons. Lagi mong tatandaan, ang math ay para sa lahat. At ako si Teacher Van, nagsasabing dito sa math isip, lamang ang nag-iisip. God bless and see you on the next video. Bye-bye.

You can check the link below or message us on our Facebook page. Alright, so let's get started with our lesson. Okay, so class, at the end of this lesson, you will be able to multiply and divide simple monomials leading to the derivation of the loss of exponent.

Okay, so let's proceed to multiplying monomials. What is our rule when it comes to multiplication? Okay, so let's have this exploration first. We will observe this example so that at the end, we are able to derive what is the rule when it comes to multiplication of monomials.

So as you can see, we have here two monomials multiplied to each other. We have here x raised to 3 or x cubed times x raised to 4 or to the power of 4. So what will be its answer? The first thing we are going to do is to expand. So the way to expand this is, since this is x raised to 3, we are going to write down 3x. Since x raised to 3 means multiply x by itself 3 times.

That's why we have here 3x. Paano naman po yung x raised to 4? Ilang x po yan?

Correct, there are 4x. Okay? Ito na po yung tinatawag nating mga factors.

Ano po ang factors? Factors are numbers or variables multiplied together to form a product. So what are we going to do with these factors?

We are going to combine. Combine the factors. So in our first monomial, x raised to 3, we have 3 factors of x.

And then x raised to 4, we have 4 factors of 4. We are just going to add them. We are going to combine them all. So how many x are there? There are 1, 2, 3, 4, 5, 6, and... So, ano po kayang magiging sagot?

Yes, it's actually x raised to 7. The question is, do the expressions have the same base? Pare-parehas po ba yung base nung dalawang monomial? Yes, ano po yung base nila? It's x. Another question, what happened to the exponents?

Ano pong nangyari sa ating mga exponents, 3 and 4, bakit siya naging 7? Okay, correct. Ang ginawa nang natin dito is we...

add the two exponents. That's why our exponent on our answer is 7. Okay, another example here, we have here x raised to 4 times y raised to 6. What will be our answer? So if we're going to do the same process earlier, we are going to expand. So we have here x raised to 4. How many factors po yung ating x dyan. There are four factors of x.

And then next one, Ilang factors po yung ating y raised to 6? Okay, ganun din po. Anim, no?

Depende sa ating exponent. So, that is the expansion. These are the factors.

Now, we are going to combine these factors. So, how do we combine these two factors? Okay, the answer here is x raised to 4, y raised to 6. May nagbago po ba sa kanila? Okay, parang pinagdikit lang po natin sila, no?

We cannot combine these two. Because they have different base. Okay?

Magkaiba po yung base nila. So we cannot just make this 10, no? Hindi natin pwedeng i-add yung 4 and then 6 to make it 10 because they have different base.

So again, do the expressions have the same base? Hindi po. Okay? What else?

What happened to the exponents? As is po yung ating mga exponents. Wala pong nangyari. We just write them. beside each other.

Wala na po yung multiplication sign. Yan na po yung mismong sagot. Okay, so basing on those two explorations, we can now derive the rule on multiplying monomials. Okay, so ano po yung rule natin? Eto po.

So we have here x raised to m times x raised to n is equal to x raised to m plus n. Meaning, if we have here a base which is x and then we have an exponent which is M And then we will multiply it with another monomial, which is x raised to n. Ang gagawin lang po natin sa mga exponents nila ay i-add lang po natin. So whether magkaiba or parehas ang kanilang exponents, we can just add the exponent. Ang tawag po natin sa rule na ito ay, product rule.

Okay? And take note, this rule only works if the bases are the same. For example, we have x or a or any number.

As long as the base is the same, we can apply this rule. If their bases are different, just like in exploration number 2, we cannot just add the exponents. Kokopyahin lang po natin. Okay? So, ito po ang product rule.

Okay, to give you an example, we have here an activity. Let's find the product of the following monomials. Okay, again, so you can always pause this video if you want to answer this on your own first. But if you want to watch first these examples, you just follow along while we answer each number. Okay, so first, number one, we have here x raised to 8 times x raised to 2. So following our product rule, let's check first kung parehas po ba ang kanilang base.

Alright? Do they have the same base? Okay, tama.

Now, we will just add the exponent. So, therefore, our answer is x raised to 10. Okay? 8 plus 2 is equal to 10. And then, kinapi lang natin yung base nya.

Next one, do they share a common base? Yes. And then, we just add 2 and 3. So, the answer is a raised to 5. Next, parehas ba ang base?

Yes. Okay, not necessarily a variable. So we'll... 2 is our base and then our exponent is 2 and 5. So we just add this, we will get 7. You can further simplify this by multiplying 2 7 times to itself so that you can get the final answer. But this can also depend on your teachers.

Next one, we have here y raised to 4 times y. So since this exponent is not there, it means it is raised to 1. Therefore, you just add 1 and then 4, so magiging y raised to 5. Again, kaya naging 5 kasi may 1 siya dito. Next, we have here 3 monomials multiplied.

So, paano ito? So, do they share a common base? Yes, they share x and then y.

Dito x and y, x and y din. So, first, let's have x. Ano ba ang exponent ng x? Exponent nya dito is 1. Exponent nya dito sa second is 2. And then exponent nya dito is 1. So you just add those 3. 1 plus 2 plus 1, it will give you 4. Next one, we have y.

Exponent nya ay 2. Dito naman ay 1. Dito naman ay 1. So, ang sagot natin, ganito. Alright. Ia-add lang natin lahat ng exponent. Therefore, our final answer is x raised to 4, y raised to 4. Okay, let's have more example. We have here number 6. We have here monomial 9x raised to 3 or 9x cubed y raised to 4. And then, multiply it to 4x squared y cubed.

So, ano pong pinagkaiba nito sa mga examples natin kanina? So, if you observe, meron na po tayong mga numerical coefficients, yung 9 and then 4. So, what do we do with this? So, just like in our exploration, we can expand this just to demonstrate.

Para lang ma-explain ko sa inyo, if we expand all the factors, we still have 9 and 4 as factors dito. So, mumultiply lang natin sila. Okay? Yung mga factors ng x cubed, nandito, 1, 2, 3. Factors ng x squared, 1 and 2. Ganun din po sa y. Alright?

So, all you have to do is to multiply 9 and 4, you will get 36. And then you combine all these factors, x raised to 5, y raised to 7. 3 plus 2 ito, kaya naging 5, and then 4 plus 3 to make it 7. So again, just multiply the numerical coefficients if there are coefficients. Okay, kailangan pa bang isulat natin itong expanded form? No, no need po, no? Sinulat ko lang po yan for demonstration.

You can go away with this and directly write the final answer. Next, number 7 we have here. We still have numerical coefficients.

Negative 5. x cubed y squared times 3, x cubed y raised to 4. So let's multiply negative 5 by 3, we will get negative 15. And then x cubed times x cubed, just add 3 plus 3, we get x raised to 6. y squared times y raised to 4, 2 plus 4 is 6. Okay, negative 15x raised to 6, y raised to 6. Okay, for our last example, we have here negative... 6 a raise to 3 b c raised to 4 times negative 3 a b c d. Okay?

So, what's the difference between these 8? So, if you observe, here in the second monomial, we already have an extra variable that's not here in the first monomial. So, what do we do with this one?

Just like that, all we have to do is to multiply their common elements, their variables. or even the numerical coefficient. So negative 6 times negative 3, we get positive 18. a raised to 3 times a is a raised to 4. b times b, since their exponent is 1, we just add 1 plus 1, you get 2. Okay, b raised to 2. Next, c raised to 4 times c raised to 1. 4 plus 1 is 5. So we have c raised to 5. And then since D has no partner here in the first monomial, you just add or you just copy our D here.

Okay? Therefore, our answer is 18, a raised to 4, b squared, c raised to 5, d. Not all bases are the same.

It doesn't need to be the same for all bases here. Okay? So those are our examples for multiplying monomials. Okay, so let's now proceed to dividing monomials.

But before that, let's have first a short commercial break for our dear teachers. Okay, so first let's explore these examples so we can also derive the rules in division of monomials. Okay, so we have here the first fraction.

As you can see, we have 5 over 5 or 5 divided by 5. So, what is the answer here? Okay, yes, correct. It is 1. Okay, in short, we can just cancel this out.

Okay, meaning it will become 1. Okay, it is equal to 1. And also with this one, negative 3 over negative 3, we just remove or cancel this out. Magiging 1. Yun po yung ginagawa natin when the numerator and denominator are the same. So, nagiging 1 lang po siya.

Hindi po, 0 ha. 1 lang. Okay? Take note of that.

So, next, what can you say about its dividend and divisor? So, where is the dividend? The one in the numerators. So we have 5, negative 3, and 10. These are all the dividends. And then the divisor are the one in the denominator.

So it's the same. We have the same numerator and denominator. So we have the same dividend and divisor.

Next, what have you noticed in the quotients of the given integer? So what is their answer? All of them are 1. Alright, so... Let's explore more on this example. We have here a variable x with an exponent of 5. And then sa denominator, a variable x with an exponent of 3. So how do we divide this?

We will expand first. Kunin natin yung mga factors. Okay, so we have here x.

Ilan po bang x ang factor nito? So there are 1, 2, 3, 4, 5. That's why it was x raised to 5. Next one, how many x dito sa denominator? Yes, 3. Okay? Now, just like in dividing those fractions, we can actually cancel out kung ano yung nasa numerator na nasa denominator din. So, we can cancel this first pair, second pair, and then third pair.

So, ano na po yung naiwan? We have 2x on the numerator. So, yun na po yung ating sagot. Since dito sa baba, ano na po ang naiwan dito sa denominator? Okay, hindi po siya zero.

Okay, may naiwan pa rin po dito na one. Hindi lang po naisulat. Okay, therefore, itong dalawang x na to ang denominator nila ay one. So just combine these factors, magiging x raised to two na po siya. Okay, so what did we observe from this?

Do the expressions have the same base? Yes, parehas po ang base. Next, what happened to... to the exponents. So, anong nangyari sa mga exponents natin?

Yung 5 tsaka 3 dito, ang naging result sa final answer ay 2. So, what did we do with 5 and 3 to make it 2? Okay, correct. We just subtracted those two exponents.

Okay? So, earlier in multiplication, we added. Now, we will subtract.

This is what we call the quotient rule. In product rule, we added exponents. Here in quotient rule, we will subtract.

It's the opposite. So if we have here x raised to m or any exponents divided by x raised to n or any exponent, we just subtract m and n. So it will be x raised to m minus n.

By the way, this rule only works if the bases are the same. Parehas din po sa product rule. Dapat parehas po ang base nila. Kung magkaiba po ang base nila, you cannot apply this rule.

Okay, another thing to take note is ito po. Pag negative yung exponent, halimbawa, We have here x raised to m and then x raised to n. Pag pinagsubtract mo sila, ang lalabas ay negative. Anong gagawin natin? Okay, yung magiging numerator natin is 1. Tapos yung variable na may negative exponent mapupunta dito sa denominator.

At ang kanyang exponent ay magiging positive. So, mamaya magbibigay po tayo ng example nito. Alright? Where m is less than n.

Okay, so we have here these examples. So again, if you want to answer this first, para lang mataray mo, matest mo, you can do so. Pero if you want to watch first these examples, panoorin nyo po muna yung gagawin natin. So number 1, we have here m raised to 5, n raised to 7, o raised to 3 over m raised to 4, n raised to 3, o raised to 3. So the first question is, pare-parehas ba ang kanilang base?

Yes, pare-parehas. Now, can we apply the quotient rule? Yes. So we have here the answer, m raised to n, I mean, mn raised to 4. Okay?

Paano po nangyari? 5 minus 4 is 1. That's why we have here m raised to 1. Do we need to write 1 here? No.

Next one, 7 minus 3. is 4. So we have here n raised to 4. Next, dito naman po sa o, 3 minus 3 is 0. Okay? So any number or variable with an exponent of 0 is equal to 1. Hindi po 0 ha, 1 lagi. So do we need to write 1 here? No need.

Okay? Kaya ito, cancel out na po yan. Pag parehas po ang exponent ng nasa numerator and denominator with the same base.

You just cancel out. Okay, dito sa number 2, okay, nabigay ko na po yung sagot. We have here 8x raised to 5 over 2x raised to 2. So, pag may mga numerical coefficient, tayo dito, we just divide. So, 8 divided by 2 is 4. And then for our x, 5 minus 2 is 3. That's why 4x raised to 3. Okay, next number 3. Ano yung magiging sagot natin? So, we have here negative 3. d f raised to 2. So paano nangyari yun?

Negative 21 divided by 7 is negative 3. Next one, d raised to 3 over d raised to 2. 3 minus 2 is 1. So you just write d. Okay, raised to 1 yan pero huwag na nating isulat yung 1. Next one, e raised to 7 and then meron ding e raised to 7 dito sa baba. You can just cancel this out.

Okay, cancel this out. Next, f raised to 5. over f raised to 3, 5 minus 3 is 2. That's why f raised to 2. Okay, number 4, we have here negative 54 over 9. Okay, pag dinivide natin ito, magiging positive 6. Okay, that's why we have here 6. Dito naman sa x natin, we have x raised to 5 over x raised to 3, 5 minus 3 is 2. Okay, and then dito, y raised to 8 over y raised to 7, 8 minus 7. is 1. That's why we have y raised to 1. We don't need to write 1. Okay? Ganun lang po siya.

Okay, let's have number 5. This time, ano po ang pinagkaiba nito sa previous examples natin? If you notice, yung nasa numerator natin, tatlo po yung variable diyan, while here in the denominator, dalawa lang. So, meron pong z na walang pares dito sa denominator. Okay, so if we divide 36 by negative 4, our answer is negative 9. Okay, next one.

We have here x raised to 3. raised to 2 here in the denominator, 3 minus 2 is 1. That's why we have here x only. Next, y raised to 3 over y raised to 3. Cancel this out. So, y is gone.

And then we just copy z here in the numerator. Okay? So, that's it.

Next, for number 6. Okay? Let's find out what we're going to do here. In these examples, from 6 to 8, you have to pay attention because this is different from our previous examples.

If you notice, the ones in the numerator have a smaller exponent as compared to the ones in the denominator. So eventually, our answer will be negative. Remember, 2 minus 5 is negative 3. And then 3 minus 4 is negative 1. So our answer is this.

x raised to negative 3. y raised to negative 1. Pero meron po ba tayong polynomial na negative ang exponents? If you go back to our previous lesson on what polynomial is? Okay, wala po. So ang gagawin natin, i-co-convert natin ito sa polynomial.

So using our quotient rule, yung nabanggit po natin kanina, na negative exponents, ang gagawin lang natin is ibababa natin ang mga ito. So, magiging fraction na po siya. Okay, and this fraction, if you notice, meron na po tayong numerator na 1 sa taas. Tapos, itong mga ito na negative ang kanilang exponents, ibinaba lang po natin sa denominator. Now, pag binaba na natin sa denominator, yung negative na exponents magiging positive na po siya.

So, we have here x raised to 3, positive 3, and then y raised to positive 1. Okay, instead of... negative 1. Okay? Ganun lang po siya. Now for number 7, we have here another example kung saan may mga variables din tayo na mas maliit ang exponents as compared dito sa baba. Take note, negative po yung ating numerator and then positive yung ating denominator.

So pag ganito po, laging ang sagot natin is negative. Okay? Negative divided by positive is negative.

So the next one is we will now subtract the exponents. So this will be its answer. Negative 3a raised to negative 1, b raised to negative 2, and then c raised to 5. Since there's no c here in the denominator, we just copy the c raised to 5 here.

Next, since we cannot have a negative exponent, we will lower the variables. So, hindi po lahat ibababa. Yung mga may negative na exponents lang po ang ibababa natin sa denominator. And then, yung numerator nya magiging 1. Pero, in this case, we have a negative 3 numerical coefficient. So, yun na po yung magiging numerator nya.

Okay? Yun na po yung magiging numerator nya. So, we have negative 3. c raised to 5, kasi ito positive naman po yung kanyang exponent, no need to bring it down, over a times b raised to 2. Okay, since itong dalawang variables na ito, sila lang po yung may negative na exponents.

Okay, for number 8, we have here negative 9 over np raised to 4. Paano po nangyari yan? We don't need to write this part, yung mga... May negative na exponent, pwede natin idiretso na sa final answer. Kung saan yung mga negative exponents natin is ilalagay na natin dito sa denominator.

Okay? So again, 72 divided by negative 8 is negative 9. Okay? Yun po ang nasa numerator natin, yung numerical coefficient. Next, yung mga variables po natin, we have here m raised to 2. m raised to 2 din po dito sa denominator. You can just cancel this out.

So, huwag na po natin idagdag sa sagot. Next, n raised to 2 over n raised to 3. Since ang sagot natin dito is negative 1, ang kanyang variable ay nasa denominator. Okay? Next, p raised to 3 over p raised to 7. Magiging negative 4 din po yung exponent dito.

So, ilalagay din po natin siya sa denominator. Okay? That's why. So, ito po lahat yung mga examples natin. I hope you understood.

You can backword if there are numbers or items that you haven't understood yet. So after this, I'm now going to show you exercises so you can also practice on your own. Okay, so this is now your activity. You find the product or quotient of the following monomials.

So part A is for the product and then part B is for the quotient. Okay, I'm going to give you a few minutes or take your time as much as you need. You can pause this video so you can do it on a separate sheet of paper. And then you can just come back later to check your answers. Okay, good luck and God bless your answer.

Okay, I hope you are now ready. I'm now going to show you the answers for this activity. Okay, so for letter A and letter B, ito na po yung mga sagot. Okay, pakicheck na po ang inyong mga answers and then I hope nakuha nyo po yung mga tamang sagot. Okay, so again, for our next lesson, we will now have multiplication and division of binomials and multinomials naman.

Okay, so I hope to see you again on the next one. Thank you for listening. God bless and bye-bye for now.

Kung meron kang natutunan sa video na ito, please click the subscribe button to support this channel. Para mas matulungan ulit kita sa mga susunod mo pang math lessons. Lagi mong tatandaan, ang math ay para sa lahat. At ako si Teacher Van, nagsasabing dito sa math isip, lamang ang nag-iisip. God bless and see you on the next video.

Bye-bye.

Transcript for:Monomials Multiplication & Division

Transcript for:
Monomials Multiplication & Division