Factor out the least power of the variable or variable express...

Question

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4x(2x+3)^-5/9+6x²(2x+3)^4/9-8x³(2x+3)^13/9

Accepted Answer

Welcome back. I am so glad you're here. We are Given this expression, we have 26 p times two P plus five raised to the negative 5/ C plus 13 P squared times two P plus five raised to the 8/13 plus 39 p cubed times two P plus five raised to the 21 13th wow. And we essentially have to factor this out. So what we're going to do to factor it is we're going to take a look at these three terms. Put the first one in red, the second one in blue and the third one in green. And we're going to compare the comparable parts. So we'll look at the whole numbers 26 13 39. These P. S P p squared P cubed and then all three have a two P plus five but they're just raised to different powers. So we'll compare those three parts of the three terms and see what we can factor out. Let's start with the whole numbers 26 and 39. What is the greatest common factor between those three terms? What is the highest number that They can all be divided by? Well, that would be 13. So because the greatest common factor is 13, we can factor out a 13. All right now let's take a look at the P. S. We've got P p squared and p cubed when we are comparing variables with exponents. This is where we are looking for the common factor with the lowest power. So we're comparing the powers of these. Or the degrees their exponents. This first one in red. That P has an unwritten power of one. So we're comparing 12 and three to find the lowest power and out of 12 and three the lowest power is one. So we'll be dividing all three of these by P. To the first or just P. So because we're putting that up here, we'll put it down here, we're factoring out the 13 P. All right now, for these two P plus fives, the first one is raised to the negative 5/13. So the second one is raised To the 8 13th And the 3rd Is raised to the 21 13th. So again, we've got variables raised to a power. We're looking for the one that is the lowest and the lowest one here is the negative one. It's the two P plus five raised to the negative 5/13. So we're going to divide again, all three of these by the factor with the lowest power. So we're dividing all three of these by two P plus five Raised to the -5 13th raised to the negative 5/13 and we'll put that down here too. So 13 p times two P plus five, raised to the negative 5/13. Alright, so we have determined what is being factored out of all three and now we need to figure out once we've factored that out what's left. So this is where we do that division that we have set up. So let's start with the first term. Everything in red we'll do 26 divided by 13 26 divided by is too So we've got two left from the first term P to the first divided by P to the first. Well whenever we are dividing variables with exponents we are going to be subtracting those exponents. So we have a one minus one in this case and one minus one is zero. So we're left with P to the zero. Well P to the zero. Anything raised to the zero power is equal to one. So this is a two times a one, two times one is two. So we don't need to put that one there. Alright and again because we're dividing variables with exponents, we are going to subtract their exponents for two P plus five, raised to the negative 5 13 divided by two P plus five raised to the negative 5/13. So I'm just going to do the work here to subtract those exponents. So we have negative 5/13 minus negative 5/13. Or in other words negative 5/13 plus 5/13 and we have a common denominator so we can combine those enumerators negative five plus five equals 0/ or just zero. So we have two P plus five and what's left for that exponents is zero. Well as we just said, anything raised to the zero power is equal to one. So again we have a two times one but two times one is two. So we don't need to put that. Alright the first one is done now. The second one let's start with the whole numbers 13 divided by 13 that equals one. So we have plus one P squared divided by P. To the first. Subtracting the exponents two minus one equals one. So we have P. Two the first or just P. So we have one times P. Here. Well one times P is P. So I'm just going to put P. And now doing the work to subtract the exponents for R. Two P plus fives we have 8/13 minus negative 5/13 which is 8 13 plus 5/13 and common denominator is there. So we add eight plus five in the numerator and that equals 13 13th which is equal to one. So we have two P plus five raised to the first which is just two P plus five. So we'll put that down here as what's left and we're done with the second one. Now let's do the 3rd 1 39 divided by 13 equals three. So we have plus three P. To the third divided by P. To the first three minus one is two. So we have P squared three P squared and for the two P plus fives will take the exponents 21 13th and subtract them minus negative 5/13 which is 21 13th plus 5/ and we can add the new writers because we have a common denominator. 21 plus five is 26 13, 26 divided by 13 is too So we've got two P plus five squared that we can put down here and we have done all of the factoring. Now what we need to do is simplify what's inside those brackets. This will be a little bit of work so we'll give ourselves some space. We're going to be bringing down this 13 p times two P plus five raised to the negative 5/13 quite a bit. And now let's do what we can so we've got to nothing we can do with the two plus we can distribute this P two, the two p plus five. So p times two P is two P squared p times five is plus five P. And now we can't distribute this three P squared yet because this two P plus five is squared and we have to do that first. So let's bring down the plus three P squared. And now we have to take two P plus five times two P plus five and we will need to foil that out. So first bringing everything down negative 5/13 times two plus two P squared plus five P plus three p squared. Ok now let's foil for our first we've got to p times two which is four P squared. Outsides to p times five is plus 10 p insides five times two P. Is plus 10 P. And last five times five is plus 25. And now we can distribute that three P squared to all of those terms in there. So again, let's bring all this down. 13 p times two P plus five raised to the negative 5/13 times two plus two P squared plus five P plus. All right now doing the distribution three P squared times four P squared is 12 P to the fourth. Three P squared times 10 P is plus P to the third. And we've got another one of those three P squared times 10 P. S again 30 P to the third and three P squared times 25 is plus 75 P squared. Okay, now it's just a matter of combining like terms we're almost done. So we've got 13 p. Times two P plus five raised to the negative 5/13 hanging out on the outside and combining our like terms, let's start with the highest degree. So that will be this 12 P to the fourth. So we've got 12 P to the fourth. Now, our next highest term is 30 P cubed plus 30 P cubed. So adding those two together is plus 60 P cubed. Now let's crp squares we've got two p squared plus 75 P squared two plus five is plus 77 P squared. And next we've got our p that's just plus five P and last are constant term. Is this too? So that'll be plus two and that is as simplified as this one gets. Now. Let's drag this back up to the top, be a mess for a second and we'll see what's our correct answer. Does it fit? Just about alright. Looking at our answer choices. We're looking for a 13 p. Times two P plus five. Raised to the negative 5/13 times 12 P. To the fourth plus 60 P. Cubed plus 77 P squared plus five P plus two. That matches with answer choice. A well done. We'll catch you on the next one.

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4x(2x+3)^-5/9+6x²(2x+3)^4/9-8x³(2x+3)^13/9

Key Concepts

Factoring Out the Least Power

Properties of Exponents

Assumption of Positive Variables

Watch next

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4x(2x+3)-5/9+6x2(2x+3)4/9-8x3(2x+3)13/9

Key Concepts

Factoring Out the Least Power

Properties of Exponents

Assumption of Positive Variables

Watch next

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4x(2x+3)^-5/9+6x²(2x+3)^4/9-8x³(2x+3)^13/9