Hi mga kawawmat! In this video, we will be discussing multiplication and division of rational algebraic expressions. So, let's have first our objectives. First, we have to recall the multiplication and division of fractions. multiply and divide rational algebraic expressions.
Okay? How do we multiply fractions? So, first, multiply both the numerators and denominators.
Example, so if I have A over B times c over d, then that will become a times c over b times d. Now, remember that your b and d should not be equal to 0 dahil kapag yan nag-equal sa 0, that will make your fractions undefined. Laging tatandaan yan.
Now, example, so if I have here 2 over 3 times 3 over 4, so 2 times 3, that is 6. 3 times 4, that is 12. And then get the lowest term, that is 1 half. Okay? Bakit natin ito nire-recall? Kasi pwede natin itong magamit sa pag-multiply ng rational algebraic expressions. Another example, if I have 4 over 5 times 2 over 4, that is 8, over 20, simplify or get the lowest term, that is 2 over 5. Now, you can also use cancellations.
Pwede rin gumamit ka ng cancellation. Mas mabilis yun. So, kung sanay kang gumamit nun, pwede rin naman.
So, that is, pwede natin i-cancel. Cancel si 2 and 4, that will become 1 and 2. And then, si 3, so mas mabilis siya ngayon. Paano, 1 times 1, that is 1, and then 1 times 2, that is 2. Okay, so pwede rin ito.
1 times 2, that is 2, and then 5 times 1, that is 5. Now, in multiplying rational algebraic expressions, so same process yan kung paano tayo mag-multiply ng fractions. Okay, kaya ni-recall muna natin siya. Okay, pwede natin siyang mag-anod.
So if I have here P over Q times R over S, so that will become P times R over Q times S. Now since ang denominator natin dito ay Q and S, dapat ang Q and S natin should not be equal to 0. And remember that your P, Q, R, and S dito are already polynomials kasi we are already in rational algebraic expression. So polynomials na siya. Okay?
So... So what are the steps in multiplying rational algebraic expressions? So first, factor the numerator and denominator. After factoring, divide or cancel out the common factors. And then, multiply the remaining terms and simplify if possible.
Let's have an example. So I have here a raised to 5 over 10 times 5 over a raised to 3. Okay, so pwede mong i-apply kung paano tayo magmultiply. multiply ng fractions. Okay? So, numerator and denominator.
So, a raised to 5 times 5, that is 5a raised to 5. And then, 10 times a raised to 3, that is 10a raised to 3. And then, simplify. So, you can get the GCF of 5 and 10 or get the lowest term of 5 and 10, that is 1 half. So, no need to write 1. Okay? And then, pagdating dito sa exponent natin, we are subtracting the exponents of the variables with the same bases.
So, kapag ang variable nyo ay pareho naman ng base, isasubtract natin ang exponent nya kapag division tayo. So, 5 minus 3, that is 2. Kaya, a squared na lang sya. Now, bakit sa taas ko sya nilagay? Kasi nga, yung exponent natin sa numerator ay mas mataas sa denominator. Okay.
Now, this is one way of multiplying rational algebraic expressions. Another way is, So, factor the numerator and denominator completely. So, therefore, we can have a cubed times a squared over 2 times 5 times 5 over a raised to 3. Okay. Bakit ko siya ginawang a raised to 3 and a raised to 2?
Okay. Kung baga, pinaghiwalay ko siya. Okay.
Ang goal natin dito is magkaroon tayo ng common factor. Since this is a raised to 3, ayan. So, para makancel ko to, ginawa ko siyang a raised to 3 times a raised to 2. And then dito naman sa...
sa 10, ginawa ko siyang 2 times 5. Para sa 5, makancel natin yung 5. Okay? And then, divide or cancel out the common factors. So, we can now cancel A raise to 3 and 5. Okay, anong natira?
Multiply the remaining terms. Since A raise to 2 na lang ang natira, so that is A raise to 2 over 2. Now, simplify if possible. Ito na yung simplified answer natin. Now, let me give you another way or shortcut. Since same base sila, i-minus natin ang kanilang exponent.
So, 5 minus 3, that is a raised to 2. So, sa numerator siya kasi mas mataas si 5. And then si 5 and 10, that will become 1 and 2. Okay, now simplify. So, a squared times 1, that is a squared. And then 2, wala nang natira dito, so that is 2. So, pwede rin ito. Okay? So, in this In this case, binigyan ko kayo ng tatlong ways on how to multiply rational algebraic expressions.
Another example. So I have here 30b squared over 6c times 4c squared over 15b raised to 4. So pwede tayong buying numerator, denominator. So 30 times 4, that is 120. b squared, c squared. And then 6 times 15, that is 90. b squared, and then c. Lagi natin i-arrange alphabetically ang ating variables.
Okay, now, so therefore, we have now 120 over 90. What is the greatest common factor? 30. Okay, so 120 divided by 30, that is 4. 90 divided by 30, that is 3. Now, ito na naman tayo sa exponent with the same basis. Okay, pwede lang tayo mag-subtract ng exponents kapag division siya at pareho sila ng base.
Tatandaan yan. So, 4 minus 2, that is 3. 2. So, b squared na lang ang matitira dyan. Saan natin ilalagay? Sa denominator kasi mas mataas si b raised to 4. So, that is b squared. And then, 2 minus 1 that is c na lang ang matitira.
No need to write 1 kapag ang exponent natin ay 1 na. Okay? So, 2 minus 1 that is c. C na lang.
Sa numerator natin siya ilalagay kasi mas mataas ito. So, c. So, therefore, our answer is 4c over 3b squared.
Another way, factor dyan yung Numerator and denominator completely. So, our goal is, kailangan magkaroon tayo ng common factor. So, since 15 to, ginawa ko tong 2 times 15. So, we can cancel out 15 later.
And then, dito sa 6 and 4, ginawa ko tong 2 times 2 at ito ay 2 times 3. Para makancel natin yung 2. And then, sa B squared, para makancel ko yung B squared, ginawa ko siyang B squared B squared. Yung B raised to 4. Okay? Now, we can divide or cancel out the common factors.
So, therefore, we have... We can now cancel 15. We can cancel B squared. We can cancel 2. We can cancel C. Okay?
Meron pa bang i-maka-cancel? Wala na. Okay? Hindi natin pwedeng i-cancel ito, si 2 at si 2. Bakit?
Pareho silang nasa numerator. So, pwede lang tayong mag-cancel kapag numerator at denominator. So, therefore, multiply the remaining terms. So, we have 2. Times 2, that is 4. And then may naiwan na isang C. And then 3 times B squared, that is 3B squared.
Okay. Another way is, kagaya ng ginawa natin kanina. So, 30B squared, ikakancel out natin dun sa 15B raised to 4. So, magiging 2, eto ay B squared. Okay.
So, 2 kasi ano siya, 30 divide 15. And then this is 1 na lang. So no need to write 1 here. Now itong b raised to 4 natin, mininus natin si 4 minus 2, that is 2. And then sa baba ko nilagay si b squared kasi mas mataas itong 4. And then dito, so magiging 3, eto ay 3, kasi 6 divide 2, 3, 4 divide 2 is 2. Now bakit si c nilagay ko sa taas kasi mas mataas ito. Eto ay only 1. So 2, yung exponent minus 1, that is 1. So eto na siya. So sa taas natin siya ilalagay.
So isod. natin, multiply 2 times 2, that is 4C. And then 3 times B squared, that is 3B squared. So, meron kayong tatlong options kung paano mag-multiply ng rational algebraic expressions. Okay, next.
Now, in this case, ang gagawin natin dito is factor the numerator and denominator completely. So, we will have 3D over 3 times F minus 4. Bakit 3? Because 3 is the common factor.
Okay, paano ba nalalaman yung common factor? Kung ano yung number na pwede mong i-divide sa kanila without remainder. So, ito, 3 times F, that is 3F.
3 times negative 4, that is negative 12. Now, dito... the common factor is 4. 4 times F, that is 4F. 4 times negative 4, that is negative 16. Now, eto naman sa denominator, diba, sabi ko nga, para makapag-cancel out tayo, ang goal natin is kailangan magkaroon ng common factor. So, para maka...
cancel out ko tong 3D, ginawa ko siyang 3D times 4D. Okay? So, now, we can now divide or cancel out the common factors.
So, we have 3D, maka-cancel na natin siya, ang F minus 4. Now, So, itong 4D na to, pwede natin siya i-expand as 4 times D, right? So, therefore, kung ito ay 4 times D, pwede rin natin makancel si 4. Okay? So, let us...
multiply the remaining terms. Now, meron pa bang naiwan sa numerator? Wala na. So, therefore, that is 1. And then, sa denominator natin, since na-cancel na yung 4 dito, so, 3 times D, that is 3D.
So, our answer is 1 over 3D. Another. Okay, so factor, denominator, and denominator completely. Now actually, dito pa lang sa given natin, yung a plus 2 raised to 3 and a plus 2 raised to 3, pwede mo na siyang i-cancel out. Okay, pero I'm sure...
showing nyo, paano sya na cancel out. Kasi, ibig sabihin ng a plus 2 raised to 3 ay tatlo. Ibig sabihin nito, tatlo din.
So, maka-cancel out sya. Okay? And then, ito, apat daw sya.
So, a plus 4 raised to 4. And then, ito. Ito only 1. So, ang makakancel lang natin dito ay isa lang. Okay?
So, we have a plus 4, a plus 4. So, ano ang natira sa numerator? Meron na lang tayong a plus 4 times a plus 4 times a plus 4 over 1. Okay? Nagmaglalagay tayo ng over 1 sa denominator. Hindi pwedeng wala yan.
Okay? So, now, we can have a plus 4 raised to 3 over 1. Diba? Divide natin. So, that is a plus 4 raised to 3. So, this will be our final answer.
Now, how do we divide fractions? Paano naman tayo mag-divide ng fractions? Okay. So, A over B divided by C over D, ang ginagawa natin, we are getting the reciprocal of our divisor.
Ano ba yung divisor? Yung second fraction natin. So, pag sinabing reciprocal, yung kabaliktaran.
So, ibig sabihin, that will become A over B and then D over C. Okay? So, ganun lang ang gagawin. Now, kapatid.
Pag nakuha na natin yung reciprocal ng second fraction natin o yung divisor, pwede na tayo mag-proceed sa multiplication. So actually, ang pag-divide ng fractions ay bumabalik lang din tayo sa multiplication. Kung paano tayo mag-multiply, ganun din sa division ng fractions. But you have to remember, your first step, get the reciprocal of your divisor.
So therefore, we will have A times D over B times C. Tatandaan, since ang denominator natin dito ay B, B, C, and D, dapat hindi siya mag-equal sa zero. Kasi magiging undefined siya. Okay, let's have an example. So, if I have here 2 over 3 divided by 4 over 5, so magiging 2 over 3 times 5 over 4. And then magiging multiplication na siya.
So, we will have 10 over 12. Get the lowest term, that is 5 over 6. Okay, bakit natin ito nire-review? Kasi pwede natin itong magamit sa pag-divide ng Rational Algebraic Express. expressions.
Okay, paano ba mag-divide ang rational algebraic expressions? Ganon din. Okay, ang pinakaiba niya lang, yung P, Q, R, and S natin are already polynomials.
Okay, so P over Q divided by R over S, so kukunin natin yung reciprocal ng ating divisor, so magiging S over R, and then proceed na tayo sa multiplication. So we are just going back to the rule, to the steps of multiplying rational algebraic expressions. So after natin makuha ang reciprocal, proceed na tayo sa multiplication. So therefore, we will have P times S over Q times R.
Remember that your P, Q, R, and S are already polynomials in rational algebraic expressions. And since ang denominator natin ay Q, R, and S, dapat itong mga to ay hindi mag-equal sa zero para hindi siya maging undefined. Let's have an example. Okay?
But before that, tingnan natin yung mga steps. So, ito yung mga steps sa multiplying rational algebraic expressions. So, pareho lang, no?
Bumabalik lang tayo sa pag-multiply ng rational algebraic expressions. Pero ang unang gagawin, kailangan makuha mo muna yung reciprocal ng divisor. So, ito lang yung nadadagdag dyan.
You have to do it first bago ka mag-proceed sa pag-multiply ng rational algebraic expressions. So, same process lang siya. Ito lang yung nadagdag na unang-una.
na mong dapat gawin. Get the reciprocal of your divisor bago ka mag-proceed sa multiplication. Let's have an example. So, I have here a squared minus 9 over a minus 3 divided by a plus 3 over 3a.
So, kunin natin ang reciprocal nito, etong divisor mo. So, we will have a squared minus 9 over a minus 3 times 3A, mapupunta siya sa taas si A plus 3 nasa baba and then proceed sa multiplication. So, babalik lang din tayo sa multiplication. So, yung divide mo magiging multiplication after mong nakuha yung reciprocal ng iyo. yung divisor.
Pag sinabi natin divisor, yung second fraction mo dito. Okay? Next. So, factor the numerator and denominator completely. So, we will have bakit sya naging a plus 3 and a minus 3?
So, this is these are the factors of a squared minus 9. Okay? Because this is what? The difference of 2 squares. So, pwede mong balikan yung video tutorial about factoring difference of 2 squares. Nandito yun sa playlist.
over a minus 3 and then 3a over a plus 3. Next, divide or cancel out the common factors. Since pwede natin makancel ito, ayan, si a plus 3, so makakancel out natin siya. Ito pwede ba?
Yes, kasi numerator and denominator naman siya. Remember na kapag nagkakancel out tayo, laging numerator and denominator. Hindi pwede pareho silang numerator at pareho silang denominator.
And then, ano na lang natira? So we only... have 3a.
Okay? Actually, it's 3a over 1. Pag sinimplify naman natin sya, 3a over 1 or 3a divided by 1, that is still 3a. Another example.
Okay, kapag meron tayong ganitong given, this is actually your numerator. Okay? This is your numerator kasi meron tayong over 1 dyan.
Pero, hindi na natin yung nilalagay. Okay? So, get the reciprocal of your divisor. So, ganun pa rin. x squared plus 2x minus 8. Times, proceed tayo sa multiplication after natin makuha ang reciprocal.
So, x squared minus 16, mapupunta siya sa numerator. At ito ay mapupunta sa denominator. Okay, next.
Factor the numerator and denominator completely. So, we will have x plus 4, x minus 2. So, this is actually factoring of trinomial. So, pwede mong balikan yung video tutorial about factoring quadratic trinomial.
Okay? And then this one. difference of 2 squares dito, so magiging x plus 4, x minus 4. And then another, this is factoring quadratic trinomial. So x minus 4, x plus 2. Okay? So we can now divide or cancel out the common factors.
What are the common factors? So we can cancel out x minus 4. Okay. Now, bakit hindi natin pwedeng i-cancel itong x plus 4 dito?
Kasi pareho silang numerators. So kagaya nga ng sinabi ko. ko, we cannot cancel expressions na parehong nasa numerator o pareho silang nasa denominator.
Okay? And then this one, baka malito kayo, baka makancel nyo to. Hindi po pwede kasi ito ay minus, this is plus.
Okay? So wala na tayong makakancel out dyan. And then now, we can now multiply the remaining terms.
So therefore we have x minus 2, lahat ng natira sa numerator ay x minus 2, and then x plus 4 squared. Bakit? kasi meron pang isa dun. So, dalawa sila.
And then, sa denominator, we only have x plus 2. So, therefore, this is our final answer. This is how we divide rational algebraic expressions. Thank you for watching this video. I hope you learned something. Don't forget to like, subscribe, and hit the notification bell para updated kayo for more video tutorials.
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