Help with radicals: find equiv. expr. for (cbrt(xy^2)*xy)/(cbrt(sqrt(x^4y^2))

[Nicki]

Hello everyone.
So i'm new to the math world and need a little help with the expression on the attachment. The question is what of the expressions is the same as the main one. I'd like some explanation too, please. Resources to learn more about this are welcome as well.
Thanks in advance!

. . . . .\(\displaystyle \large{ \dfrac{\sqrt[3]{\strut xy^2\,}\, \times\, xy}{\sqrt[3]{\strut\, \sqrt{\strut x^4y^2\,}\,}} }\)

\(\displaystyle \mbox{Which of the following is equivalent to the expression given above?}\)

\(\displaystyle \large{ \mbox{(A) }\, \sqrt[4]{\strut (xy)^3\,} }\). . . . .\(\displaystyle \large{ \mbox{(B) }\, y\, \sqrt[3]{\strut x^2 y\,} }\). . . . .\(\displaystyle \large{ \mbox{(C) }\, \sqrt[4]{\strut x^2 y^3\,} }\). . . . .\(\displaystyle \large{ \mbox{(D) }\, \sqrt{\strut x^3\,}\, \times\, \sqrt[4]{\strut y^3\,} }\)

 

[stapel]

...what of the expressions is the same as the main one. I'd like some explanation too, please. Resources to learn more about this are welcome as well.

Unfortunately, it is not reasonably feasible to attempt within this environment to teach you the chapter (or two?) on radicals. To learn about radicals and simplifying their expressions, try using a search engine, such as Google.

. . . . .\(\displaystyle \large{ \dfrac{\sqrt[3]{\strut xy^2\,}\, \times\, xy}{\sqrt[3]{\strut\, \sqrt{\strut x^4y^2\,}\,}} }\)

\(\displaystyle \mbox{Which of the following is equivalent to the expression given above?}\)

\(\displaystyle \large{ \mbox{(A) }\, \sqrt[4]{\strut (xy)^3\,} }\). . . . .\(\displaystyle \large{ \mbox{(B) }\, y\, \sqrt[3]{\strut x^2 y\,} }\). . . . .\(\displaystyle \large{ \mbox{(C) }\, \sqrt[4]{\strut x^2 y^3\,} }\). . . . .\(\displaystyle \large{ \mbox{(D) }\, \sqrt{\strut x^3\,}\, \times\, \sqrt[4]{\strut y^3\,} }\)

Once you have studied at least three lessons from the list at the link, please attempt the exercise. If you get stuck, you can then reply with a clear listing of your steps, starting perhaps with the following:

. . . . .\(\displaystyle \large{ \dfrac{\sqrt[3]{\strut xy^2\,}\, \times\, xy}{\sqrt[3]{\strut\, \sqrt{\strut x^4y^2\,}\,}} }\)

. . . . . . . .\(\displaystyle \large{ =\, \dfrac{\sqrt[3]{\strut xy^2\,}\, \sqrt[3]{\strut x^3 y^3\,}}{\sqrt[3]{\strut\, x^2 y\,}} }\)

. . . . . . . .\(\displaystyle \large{ =\, \dfrac{\sqrt[3]{\strut (xy^2)\, (x^3 y^2)\,}}{\sqrt[3]{\strut\, x^2 y\,}} }\)

...and so forth. Thank you! :wink:

 

[Otis]

So i'm new to the math world

\(\displaystyle \large{ \dfrac{\sqrt[3]{\strut xy^2\,}\, \times\, xy}{\sqrt[3]{\strut\, \sqrt{\strut x^4y^2\,}\,}} }\)

Great! Are you currently enrolled in a math class? What have you tried or thought about, thus far?

The answer to the posted exercise may be obtained by at least three different methods, that I know of.

On exercises like this one, my preference is to switch from radical notation to exponential notation, followed by using properties of exponents to begin simplifying (and also to eliminate the ratio), and then switching back to radical notation at the end for a final simplification.

That's a lot of instruction to cover; at this site, we generally provide guidance instead of classroom material. I'll wait to see if you post any work or thoughts, before typing out a similar exercise as an example, as I don't know whether you're familiar with fractional exponents and their properties.

In the meantime, here are some resources for simplifying radical expressions.

http://www.purplemath.com/modules/radicals.htm

https://www.google.com/webhp?#q=simplifying radical expressions

:cool:

 

[Ishuda]

Hello everyone.
So i'm new to the math world and need a little help with the expression on the attachment. The question is what of the expressions is the same as the main one. I'd like some explanation too, please. Resources to learn more about this are welcome as well.
Thanks in advance!

. . . . .\(\displaystyle \large{ \dfrac{\sqrt[3]{\strut xy^2\,}\, \times\, xy}{\sqrt[3]{\strut\, \sqrt{\strut x^4y^2\,}\,}} }\)

\(\displaystyle \mbox{Which of the following is equivalent to the expression given above?}\)

\(\displaystyle \large{ \mbox{(A) }\, \sqrt[4]{\strut (xy)^3\,} }\). . . . .\(\displaystyle \large{ \mbox{(B) }\, y\, \sqrt[3]{\strut x^2 y\,} }\). . . . .\(\displaystyle \large{ \mbox{(C) }\, \sqrt[4]{\strut x^2 y^3\,} }\). . . . .\(\displaystyle \large{ \mbox{(D) }\, \sqrt{\strut x^3\,}\, \times\, \sqrt[4]{\strut y^3\,} }\)

I would start by working on the numerator and denominator separately and turning all radicals into exponents, i.e.
.\(\displaystyle \large{\sqrt[3]{\strut xy^2\,}}\, =\, x^{\frac{1}{3}}\, y^{\frac{2}{3}}\)