In Exercises 83–90, perform the indicated operation or operations...

Question

In Exercises 83–90, perform the indicated operation or operations. (2x−7)^5/(2x−7)^3

Accepted Answer

Hello. Today we're going to be simplifying algebraic expression. So the expression given to us is the quantity five X plus seven raised to the fifth. Power over five X plus seven raised to the second power. Now this matches the form of exponential division. Where you have X. To the A. Power over X. To the B. Power equal to X. The power of A minus B. Here we're gonna let X equal to five X plus seven. And we can rewrite the exponential expression as five X plus seven. Race in the power of five minus two and five minus two is just going to be three. So what we end up with is the quantity five X plus seven raised to the power of three. Okay now we're gonna need to expand this quantity while five X plus seven race to the third. Power is five X plus seven times itself three times. So we can rewrite this as five X plus seven times five X plus seven times five X plus seven. And in order to expand this we're gonna need to foil each quantity with each other. Well we're gonna need to foil twice because if I group up the first two mono me als I can foil those and then I can foil the last five X plus seven into its product. So let's start by foiling the first two quantities of five X plus seven together. So again we're gonna foil five X plus seven with five X plus seven. What we're going to do here is take the first term five X. And multiply to the 1st and 2nd term of the second quantity. Then we're taking our second term and repeating that process of multiplying to the 1st and 2nd term of the second quantity. So five X times five X. Is going to give us 25 X squared five X times seven is going to give us 35 X seven times five X. Is going to give us 35 X. And seven times seven is going to give us 49. All right now, what we need to do is go ahead and combine the like terms together. While the only things that we can combine our both the values of X. So combining those together is going to give us the try no meal 25 X squared plus 70 X plus 49. So this is the product of five X plus seven times five X plus seven. But keep in mind that we still have this third quantity five X plus seven that we need to multiply with this product. So rewriting this is going to give me five X plus seven times 25 X squared plus 70 X plus 49. And we're going to need to foil this together. And it's essentially adding the same another step to the foiling process where we take our first term five X. And we're going to multiply to the 1st, 2nd and 3rd term of the binomial. Then take our second term seven. And repeat that process by multiplying to the 1st, 2nd and 3rd term of the binomial. So let's go ahead and start that five x times 25 X squared is going to give me 125 X cubed five X times 70 X. Is going to give me 350 X squared five x times 49 is going to give me 245 X. Now seven times 25 X squared is going to give me 175 X squared Seven times 70 is going to give me 490 x And seven times 49 is going to give me 343. Now what we need to do is go ahead and combine the like terms. The only combining like terms here is going to be 350 X squared with 175 X squared because they both have the X squared term and 245 X with 490 X. Because they both have an X term combining all these quantities together is going to give me the polynomial. 125 X cubed plus 525 X squared plus 735 X plus 343. And that's gonna be the simplified version of this algebraic algebraic expression. And that means that the answer to this problem is going to be deep. So I hope this video helps you understanding how to simplify this algebraic expression, and I'll go ahead and see you all in the next video.

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