In Exercises 69–82, multiply using the rule for the product of th...

Question

In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.

(y³ + 3)(y³ − 3)

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to perform multiplication on the given expression. The expression we're given is two binomials multiplied together. So we have the first H to the exponent 4 plus a multiplied by the 2nd Binomial H to the Exponent 4 -8. We're given four answer choices. Option A H to the exponent eight plus 64. Option B H to the exponent eight minus 16, H plus 64. Option C H to the Exponent 8 -64. And option DH to the exponent eight plus 16 H plus 64. We're gonna start by rewriting the expression we were given. So we have the binomial H to the exponent four plus eight multiplied by the Bino binomial H to the exponent four minus eight. If we look at this, we can recognize that this is a difference of squares. OK. Recall for a difference of squares. If we have the binomial A plus B multiplied by the binomial A minus B, we can write this as the difference of squares A squared minus B squared. So if we have what's on the left hand side and it gives us a nice, quick, easy way to expand the two factors. OK. So in our case, this is exactly what we have. We have H to the exponent four plus eight multiplied by H to the exponent four minus eight. So we have the left-hand side of our difference of squares expression where A is equal to H to the exponent four N B is equal to eight. So now our original expression eight to the exponent four plus eight multiplied by H to the exponent 4 -8 can be written as A squared. Our A term is H to the exponent four. So we have H to the exponent four squared minus B squared. So minus eight squared and now this expression has been expanded and all we need to do is simplify. So we have H to the exponent four squared. Can you recall that when we have an exponent raised to an exponent, we have X to the exponent M raise to the exponent N, you can write this as X to the exponent M multiplied by N. OK. So if we have an exponent to an exponent, we can multiply the two exponents together to combine them into one. And so we can write H to the exponent four squared as H to the exponent eight -8 Squared, which we know is 64. And so this is the result of the multiplication. This is our simplified expression that we were looking for. Now, one comment here, if you didn't recognize that this was a difference of squares, that's OK. OK. We have two binomials multiplied together and there are plenty of other ways to multiply them. OK. You could use the foil method. For example, if you work that out, you're gonna get the exact same result. The difference of squares is great because it gives you a more efficient way to do it. You don't have as many intermediate steps where you can mix up spines and things like that. Um So it's really great if you recognize it, but if not go ahead and do that multiplication and you'll get the exact same result. And that result that we found is H to the exponent 8 -64, which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms. (y³ + 3)(y³ − 3)

Key Concepts

Product of Sum and Difference

Polynomial Multiplication

Difference of Squares

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