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

Question

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 2(3x+1)^-3/2+4(3x+1)^-1/2+6(3x+1)^1/2

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression three times five P plus two, raised to the negative 2/5 plus nine times five P plus two raised to the 3/5 plus 12 times five P plus two raised to the 8/5. And essentially we need to factor this. So what we're going to do is look at these three terms, I'll break them down. The first in red, second in blue and the third in green. And we are going to compare the comparable parts. So we'll look at the whole numbers 39 and 12 and then they all have this five P plus two but they're raised to different degrees. And so we're going to compare each of those and see what we can factor out. Let's start with the whole numbers, we've got three, nine and 12 and will divide each of these by the greatest common factor. So what is the greatest common factor for? Three? Nine and 12? While all three are divisible by three. So we're going to factor out a three right now, let's look at these. Five P plus twos. The first one is raised to the negative 2/5. The second one is five P plus two, raised to the positive 3/5 and the third one is five P plus two raised to the 8/5. So what's the greatest common factor for those? Well, we are going to be now looking for the common factor with the lowest power. So we're going to be looking at their degrees, their powers for all three of these because they're all five P plus two but they have different powers and we're going to look for the lowest one and in this case the lowest one is the negative 2/5. So we're going to divide all three of these by five P. Plus two, raised to the negative to fifth. So five P plus two to the negative 2/ and five P plus two raised to the negative 2/5. And we will put that down here because we are factoring it out five P plus two raised to the negative 2/5. So now that we factor that out now we ask ourselves well what's left after? We do all of that division. So now we'll do the division. Let's start with the first term. So doing the red, we'll take the three divided by three. Well three divided by three is one and now we'll do the five p plus two, raised to the negative 2/5 divided by five P plus two, raised to the negative 2/5. Well when we are when we have when we have variables, excuse me, the five P plus twos and they are raised to exponents. And being divided, we are going to subtract those exponents. So we'll take negative 2/5 minus negative 2/5. Or in other words we're taking negative 2/5 plus 2/5. We've got a common denominator. So we can add the numerator, negative two plus two is zero. So we have 0/5 which is zero. So this five P plus two is raised to the zero power. Well anything raised to the zero is equal to one. So down here we have a one times one and one times one is just one. So I won't write that second one. All right. The first term has been factored. Now let's do the nine times five P plus two raised to the 3/ for the nine we've got nine divided by three and nine divided by three is three. So plus three. And for the five P plus twos we're going to again subtract the variables or the exponents. So we've got 3/5 minus negative 2/5 which is 3/5 plus 2/5 common denominators. So we add the numerator is three plus two is five. So 5/5 5 5th equals one. So we've got five P plus two raised to the first power which is five P plus two and we'll bring that down here. Five P plus to write the second one is done now we'll do the third one, 12 divided by three is four. So we've got plus four and again we will subtract our exponents 8/5 minus negative 2/5. Or in other words 8/5 plus 2/ common denominators. So we add the numerator eight plus two is 10. So 15th which is equal to two. Alright this exponent is two. So we've got five P plus two squared. That's being multiplied by this. Four down here and now we've done the factoring now, it's just a matter of simplifying what is left inside the brackets and this takes a little bit of work so we'll give ourselves some space bringing down this three times five P plus two will just be bringing that down quite a lot raised to the negative 2/5. Now let's do what we can inside the brackets will bring down the one plus. Now we can distribute this three to the five P plus two. So three times five P. S 15 P plus three times two is six and we can't distribute this plus four because the five P plus two over here is squared. So we'll bring down plus four and now we've got to take five P plus two times five P. Plus two. And on the next line we can foil that out. So bringing down our three times five P plus two, raised to the negative 2/5 times one plus P plus six plus four times. Alright now, foil. First five p times five p. Is 25 P squared. Outsides five p times two is plus 10 p insides, two times five ps plus 10 P. And last two times two is plus four. Now we can distribute this four. So bringing down three times five p plus two raised to the negative 2/5 time's not going to combine like terms yet. I'm going to distribute the four first. Okay, four times 25 P squared is plus 100 p squared four times 10 ps plus 40 P. And again four times 10 ps plus 40 P. And four times 4 16. So plus 16. Okay, now let's combine like terms will bring down three times five P plus two, raised to the negative 2/5 and let's look for the highest degree term first. This is that 100 P squared. So that can go first. Now let's look for rps. We've got 15 P plus 40 P plus 40 P. That's plus 95 P. And our constant terms. We've got one plus six plus 16. That is plus 23. Alright, We've got that inside combined. We've got this simplified. Let's drag this up. Be a mess for a second. Let's find our matching answer. Alright, so we're looking for three times five P. Plus two, raised to the negative 2/5 times 100 P squared plus 95 P plus 23. This matches with answer choice C. Well done. We'll catch you on the next one

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 2(3x+1)^-3/2+4(3x+1)^-1/2+6(3x+1)^1/2