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.

$7 (5 t + 3)^{- \frac{5}{3}} + 14 (5 t + 3)^{- \frac{2}{3}} - 21 (5 t + 3)^{\frac{1}{3}}$

Accepted Answer

Welcome back. I am so glad you're here. We are given the expression 13 times two. W Plus seven raised to the negative 7/ plus 26 times two W Plus seven raised to the negative 3/4 minus 52 times two, W Plus seven raised to the 1/4. And we essentially need to factor this. So we're going to look at these three terms. I'm going to color code them, we've got the red, The blue and the green. And we're going to look at the corresponding parts of each of the terms. So we've got these whole numbers in front 1326, negative 52 and then they all have a two W plus seven but they're raised to different degrees. And so we're going to compare them and see what we can factor out. Let's start with the 13 and -52. What's the greatest common factor for those three numbers, what can we divide out of all three of those? And they're all divisible by 13. So that means we can factor out a 13. Alright now let's take a look at these two W plus seven's. We've got two W plus seven raised to the negative 7/4. We've got two W plus seven raised to the negative 3/4 and we've got two W plus seven raised to the positive 1/4. Now what we are doing is trying to find the common factor and here's where we do this. Common factor with the lowest power when we have variables rate and they're all the same and their race to a degree or a power, we are going to divide them all just like we divided the whole numbers, but we're going to divide them all by the one with the lowest power. So in this case we look at negative 7/4 negative 3/ and 1/4 and ask ourselves what's the lowest power? Well, the lowest power here is the most negative number, the one closest toward negative infinity, which is negative 7/4. So we'll divide all of these by two W plus seven raised to the negative 7/4. And that is our common factor. Oops, that's not a for our common factor with the lowest power that we are pulling out negative 7/4 and we put that down here too. So 13 times two W plus seven raised to the negative 7/4. And now that we figured out what we can factor out now, we need to figure out what's left now that we factor that out and here's where we do that division. So let's start with the first term. Everything in red will take 13 divided by 13, 13 divided by 13 equals one. And now we'll do two W plus seven, raised to the negative 7/4 divided by two, W plus seven raised to the negative 7/4. Well, when we are dividing variables that are the same and they have exponents, we are going to be subtracting their exponents. So we'll be taking negative 7/4 minus negative 7/4. Or in other words this is negative 7/4 plus 7/4. And so the denominators are the same. We can add across the numerator, negative seven plus seven is zero. So this is 0/4 or zero. So our exponent is zero. We have two W plus seven raised to the zero power. Well anything raised to the zero power is equal to one. So we have one times one down here. Well one times one is one. So we don't need to put that second one. Alright, the first one is done. Now let's work on the second term. All the blue 26 divided by 13 equals a positive two. So we have plus two and now let's do two W plus seven raised to the negative 3/4. So again we are going to subtract our exponents because the bases are the same. So negative 3/4 minus negative 7/ or in other words negative 3/4 plus 7/4. Denominators are the same, so negative three plus seven is a positive four. So we have 4/4 which is one. So we've got two W. Plus seven raised to the first which is just two W plus seven. So that's down here being multiplied by that too. All right. The second one is done. We have factored out of the first two terms. Now let's do the third one. The green negative 52 divided by 13 is negative four. So we have minus four down here and again we are going to subtract our exponents, we have 1/4 minus negative 7/4 or 1/4 plus 7/4. Denominators are the same one plus seven in the numerator, xyz 8/4 which is equal to two. So we've got two W plus seven squared and that's being multiplied by this negative forward down here. Alright, that's what's left after. We've done our factoring now, what we need to do is simplify what's in the brackets. That's going to take a little bit of work. So we'll drag this down. We've got this 13 times two W. Plus seven raised to the negative 7/4 we don't need to do anything with that. So we're going to bring that down a bunch of times. And now let's do what we can inside the brackets, bring down the one, nothing we can do with that yet. Plus let's distribute this two to the two W Plus seven, two times two W. Is four. W. Two times seven is plus 14. Now we can't distribute this negative four yet because this two W plus seven is squared, so we'll bring down minus four and that will be multiplied by two W. Plus seven times two W. Plus seven. And on the next line we can foil that out. So bringing everything down. Native 7/4 times one plus four W plus 14 minus four times. Alright, let's foil this first two W times two W. Is four W squared. Outsides two W times seven is plus 14 W. Inside seven times two W Is plus 14. W last seven times seven is plus 49. Now we can distribute this negative four, so bringing it all down plus seven times raised to the negative 7/4. Ok, I'm not going to combine like terms yet. I'm going to do the distribution first. So negative four times four W squared is minus 16 W squared, negative four times 14 W is negative W. Both times you have another negative four times 14 W. And the negative four times 49 is minus 196. Okay, now we can combine like terms, let's bring down our 13 times two W plus seven. Remember when we factor that raised to the negative 7/4 OK, Let's start with the highest degree terms. We've got this -16 W squared. So we'll start with that minus 16 W squared. Now, let's find all of our ws. We have four W minus 56 W minus W. That will leave us with a minus 108 W. Now let's do our constance one plus 14 minus 196 that will leave us with minus 181. All right. Now, we have simplified what's inside. If we were to bring this back up to the top, we're not actually going to find any answers that match that, because they have ensured that the highest degree term inside the brackets here, this negative 16 W squared is positive. And the way they did that is they factored out a negative one, that's positive one. They divided it by negative one or all three terms what you do to one you've got to do to the others. So they factor that out. So in front they've got negative one times 13 which is negative 13 times two W plus seven raised to the negative 7/4. And then inside here it's negative 16 divided by negative one, which is a positive 16 W squared and then negative + divided by negative one is a positive +108 W. And negative 1 81 divided by negative one is plus 81. And that is as simplified as we're going to get this one, see if I can bring it up, it'll be a mess for a second and looking for our matching answer choice. Right? So we've got negative 13 times two W plus seven raised to the negative 7/4 times 16 W squared plus 108 W plus 181. This matches with answer choice. D 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.
$7 (5 t + 3)^{- \frac{5}{3}} + 14 (5 t + 3)^{- \frac{2}{3}} - 21 (5 t + 3)^{\frac{1}{3}}$

Key Concepts

Factoring Expressions with Variable Exponents

Properties of Exponents

Assumption of Positive Variables

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