In Exercises 15–58, find each product. (1−y^5)(1+y^5)

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Accepted Answer

Hi there. So this problem says determine the product of the given expression. 81 minus X to the fourth times 81 plus X to the fourth. Let's go ahead and rewrite this just real quick, 81 minus X to the fourth. Multiplied by 81 plus X to the fourth. No to multiply these together we will make use of what's called foil technique. So the foil technique space that we have our first terms, our outer terms, our inner terms and are less terms. So the first terms and multiplication are just the first terms of our two expressions. So if we look at our expressions we have 81 X to the 4th, 81 plus X to the fourth. The first terms are the terms that appear first in these expressions. So our 81 And our other 81 gonna multiply those together 81 times 81. Now the other terms are the terms on the outside of both of our expressions. So we look at this as a whole. The outside terms would be 81 is the farthest smokes to the left and our positive X to the four which is our farthest most to the rent. So you multiply those two together. So you have plus 81 times x to the fourth. Now we have the inner terms similar to the other terms. The inner terms are just the two terms that our innermost and our set of expressions. So in this case you have negative X to the 4th and 81. So a minus X to the fourth times 81. Finally, we have our last terms for our last terms are the last terms that appear in both expressions. Sophie look here negative X to the fourth appears last in this expression and positive to the fourth appears last. And the other expression So we have negative X to the fourth times X to the fourth. Now we can go ahead and simplify our expressions First. We have 81 times 81 which is 65, We have 81 times X to the fourth. Simplify that. That is just 81 X to the fourth. We have X to the 4th negative X to the fourth letter times 81. That concerns the -81 X to the 4th. Finally we have negative X to the fourth times X to the fourth. That gives us negative X to the eighth. Now with this X. The fourth times X to the fourth, we're multiplying two expressions with exponents. We multiply two expressions with exponents. Let's say we have X. M times X to the N. That we just add the X funds together. Cm plus in. So that's why here we did X. The fourth times X to the fourth. That is just X. The four plus four which is X to the eighth. So we have a negative X. V eight there. Now that we have this we can combine our like terms. So if you look here you have 65 negative X. The eighth which don't have any more like terms. However we do have 81 X to the fourth and negative 81 X to the fourth. Since those are like terms we will add those together Life term just means they have the same exponent in this case. So let's add those together. You have 81 X to the 4th -81 next to the fourth. Those are actually going to cancel each other out. So Those cancel each other out. All we have left is just 65, -X to the 8th. So the answer is 65 61 -1 to the 8th which is answer a. Alright I hope to help you solve the problem. Thank you for watching. Goodbye.

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