Perform the indicated operations and/or simplify each expression....

Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √(x⁵y³)/z²

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, we have the square root of this whole thing in the numerator we have X to the seventh, Y to the fifth. And then in our denominator we have Z squared. Alright. We recall from previous lessons that this can be rewritten due to the quotient rule of radicals as taking the square root of our numerator. So the squared of X to the seventh, Y to the fifth and taking the square root of our denominator which is Z squared. And now we can concentrate on simplifying in the numerator and simplifying in the denominator. Well in order for anything to come out of this radical, there needs to be two of them due to this invisible index of two here in our radical. And so we can break apart our X. To the seventh and our Y to the fifth so that we have groups of two, so X to the seventh. That would be X squared times X squared. There's four out of the seven and then there's another times X squared. There's six out of our seven. And then there's another just an X to the first. And that gives us our seven. X. Is same with why we can break Y down to a Y squared times Y squared. There's four of those wise. And then there's one why leftover that gets us to our five wise. And yes this radical needs to be much bigger. Okay, there's all of our exes and wise in the numerator and the denominator. Well we just have two Z's. So that one's all set and ready to go. Now let's count how many can come out of the radical in the numerator. This is 123 X squared squared. That get to go through the radical and come outside. So there's three of them. So we have X to the third, that's outside. Now how many of our Y squared get to come out? 12 of them get to come out. So that will give us a Y squared on the outside. There's two of them and now that's still times what's left inside the radical which is this one. X. And this one Y. So X. And Y are left on the inside. And then as we said, our Z squared. That gets to come out. And so that's just one C. That gets to come out. And because our X. And our Y and our Z have no factors in common, there isn't any more simplifying that we can do. This is our final answer. So we're looking for an answer choice that has in the numerator, X cubed y squared times the square root of X. Y. All over the denominator. Just Z. And that matches with answer choice B. Well done. We'll catch you on the next one

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √(x⁵y³)/z²

Key Concepts

Radical Expressions

Properties of Exponents

Simplifying Fractions

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