Find each product. Assume all variables represent positive real n...

Question

Find each product. Assume all variables represent positive real numbers. (p^1/2-p^-1/2)(p^1/2+p^-1/2)

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, parentheses to be to the 1/ plus B to the negative 1/ times another set of parentheses to be to the 1/8 minus b to the negative 1/8. And we are asked to find the product or to multiply these two sets of parentheses by each other. Well normally we would go through the foil process. We look at our first, then our Outsides or insides in our last. But look the first term in each set of parentheses is the same. To be to the 1/8. And the second term in each set of parentheses is the same. B to the negative 1/8. The only thing that changes is this plus and this minus in the middle. And we see this and we recognize this is a special product, this is parentheses X plus Y times parentheses X minus y. And so we can use a special product rule because we know X plus Y times x minus Y equals x squared minus y squared. We can identify our X and our Y. And then plug it into the right hand side of the equation, skipping that whole foil process. So what's R X? R X is the first term in each set of parentheses X equals two B to the 1/8. And what's our why are why is the second term in each set of parentheses Y equals B to the negative 1/8. So now we can plug those in on the right hand side. So we have to be to the 1/8 for our X. And that is squared minus B to the negative 1 8th for our Y. And that also is squared. So now we just have to simplify, we recall from previous lessons that when we have an exponent raised to an exponent we are going to be multiplying those exponents. And then we also do need to remember to square this to this coefficient to so two squared is four. And then we have B to the 1/8 squared. So we're multiplying 1/8 times 2. 1/8 times two is 2/8 will simplify that on the next step then minus. We have B to the negative 1/8 squared. So negative 1/8 times two is B to the negative 2/8. Now we can simplify those fractions in our exponents. We've got four B to the 2/8 is 1/ minus. Well we've got a negative in our fraction here. So that means our B to the 2/8 is going down in the denominator. So it's one over B to the 2/8 and 2/8 simplifies to 1/4 which is positive now because it's in the denominator. So B to the 1/4. So we look at our answer choices, we're looking for four B to the 1/4 minus one over B to the 1/4 which matches answer choice, B. Well done. We'll catch you on the next one

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Find each product. Assume all variables represent positive real numbers. (p^1/2-p^-1/2)(p^1/2+p^-1/2)

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