Perform the indicated operations. Assume all variables represent ...

Question

Perform the indicated operations. Assume all variables represent positive real numbers. ∛64xy² + ∛27x⁴y⁵

Accepted Answer

Hello everyone. We're going to simplify the expression. The third root of 125. H squared K. Plus the third root of 729. H. To the fifth K. To the fourth. I'm going to simplify each term independently and then simplify the expression as a whole. The first term I'm simplifying is the third root of 125. H squared K. I'm gonna separate this into constants and variables and treat them each a little differently. So for my constance which is the 125, I'm going to do the third root of 125 and then for my variables, I'm going to change the radical into an exponent. So I get H. Two the one I'm sorry each to the second power to the one third times K. To the one third, The 3rd root of 125 is five. And then when we raise the power to power we multiply our exponents. So we get H. Two the two thirds and K. To the one third. Doesn't have anything to simplify. So it's K. To the one third. Now that there I'm looking at it, there's nothing else to simplify in this term. Since all of our answers are in radical form, we're gonna move this back into radical form. The five no longer needs the radical because we took care of it. And since the denominators of both of our exponents are the same. We know we are still under the same radical. We're under the third route. So the third root of H squared K. So our first term simplified to five times the third root of H squared K. Our next term which is the third root of 729. H. To the fifth K. The fourth. We're gonna do a lot of this similar stuff, Same steps. So I'm gonna take my constant and I'm gonna do the third root of 729. And then for my variables I want to treat them with an exponent. So H. To the fifth to the one third Times K. to the 4th to the 1/3. The Third Root of 729 is nine. And then again raising a power to a power H. The fifth to the one third is H. Two the five thirds and then K. To the fourth to the one third is K. Two the four thirds. So unlike our previous term there's one more thing we want to do before we bring this back into radical form and that's because I have improper fractions as my exponents. This means I can factor out a value of one and simplify it. So the nine stays put. I'm gonna just leave him out front. I can break the H down so I'm gonna get H. Two the three thirds because three thirds is one times H. To the two thirds and same thing with K. I can do a K. Two the three thirds times K. To the one third. So simplifying anything that I just made to the three thirds into the first. Power means I now have nine H. To the first which is H. Times H. two. The 2/3 times K. Two the first which is K. Times K. two The 1 3rd. I'm gonna take anything that doesn't have an exponent and pull it to the front. So I have nine H. K. Times H. To the two thirds times K. To the one third. Now that I don't see anything else I can simplify now I will put it back into that radical form. So again things without exponents go outside of the radical. The things with the exponents since they have the same denominator are both going under an exponent or sorry under a radical with an exponent of three. So it's gonna third index. So we get H squared K. To the first or just K. And now that we've simplified both of them we're going to simplify the expression. So I'm gonna go back to red And we have five times the third root of H squared k minus nine H. K. Times the third root of H squared K. So looking at this hopefully you notice that they have the same radical. They both have the third root of H squared K. This means we can factor it out. So we're gonna have five - H. K'm Gonna put that in parentheses times the third root of H squared K. This is as simplified as it gets, and this would be answer choice B. Have a good day.

Perform the indicated operations. Assume all variables represent positive real numbers. ∛64xy² + ∛27x⁴y⁵

Key Concepts

Radical Expressions

Properties of Exponents

Combining Like Terms

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