In Exercises 111–114, simplify each expression. Assume that all v...

Question

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (49x^−2y^4)^−1/2(xy^1/2)

Accepted Answer

Welcome back. I'm so glad you're here. We've got an expression we get to simplify it and we're told that all variables are greater than zero which means that we don't have to worry about domain for any of these got 27 X to the negative two, Y to the sixth. All raised to the negative one third. And that's being multiplied by X times Y to the three halves. So there are many different ways to begin to solve this and the way that I'm taking might not be what you have done but we can take different routes and end up at the same answer. I'm going to take this negative 1/3 exponent and I'm going to distribute it to these three terms. And we recall from previous lessons that when we have an exponent raised to another exponent then that exponent will be multiplied by the one it's being raised to so 27 that's like 27 to the first. So we're going to take the first that isn't visible times this negative one third. So now it's 27 to the negative one third. That X is currently raised to the negative two and we're going to take that negative two times negative one third Y is raised to the sixth. We're going to take that sixth and multiply it by negative one third. Now we've distributed distributed that negative one third and those parentheses are gone. We still bring down this X. And this Y to the three halves. Now we can go ahead and do the multiplication. So we've got 27 to the negative one third X. Alright negative two times negative one third is going to be X to the negative X to the positive because the negative times a negative X to the positive two thirds. We've got why raised the six times negative one third. Well that's going to be six times negative one third equals negative six thirds which equals negative two. So why did the negative two times X. And times Y to the three halves. And now for my next step I'm going to combine like terms. So we've got X to the two thirds times X. And we recall that when we've got um base terms X times X. That are the same and they've both got exponents. We're going to add the exponents together. So this will be X to the two thirds plus one because there is an invisible one here that goes with this X. And now let's combine our Y terms. So we've got Y raised to the negative two plus three halves and now we can go ahead and do that. Addition 27 to the negative one third is brought down and X. Alright we've got two thirds +11 is three thirds. So two thirds plus three thirds will give us five thirds and why we've got negative two plus three halves while negative two is negative four halves. So negative four halves plus three halves gives us negative one half. Now we've got them all consolidated what I am going to do next is take a look at these negatives and the negatives tell us that the terms associated with them have to go to the other side of our division. There isn't any division happening yet. Now we get to bring these 27 to the now it's to the positive one third to the denominator and this why? Now? It's to the positive one half down to the denominator. Great, those negative exponents are gone and our X to the five thirds remains triumphant. Here in this numerator Can do a little bit more simplifying this 27 to the one third in the denominator while that's like saying the cubed root of 27 which is three. And I'm looking at our answer choices and they don't want fractions exponents, They want them written as cubed in square roots. So continuing with our denominator. Why to the one half is the same thing as the square root of why. And in our numerator, X to the five thirds that is the same as the cubed root of X to the fifth. We look at our answer choices and this matches with answer choice. C. Well done. We'll catch you on the next one

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