Simplify 5th-rt(27x^2 / y^3) * 5th-rt(9x^6 / y^4)

[Math325]

\(\displaystyle \sqrt[5]{\strut \dfrac{27x^2}{y^3}\,}\, \cdot\, \sqrt[5]{\strut \dfrac{9x^6}{y^4}\,}\)

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both are fifth roots

 

[stapel]

\(\displaystyle \mbox{Simplify }\, \sqrt[5]{\strut \dfrac{27x^2}{y^3}\,}\, \cdot\, \sqrt[5]{\strut \dfrac{9x^6}{y^4}\,}\)

I have retrieved your remote image (the one that was legible), typed out the exercise, and moved your question to an appropriate category. Now please reply with a clear listing of your thoughts and efforts so far, so we can see where you're getting stuck. For instance, you used what you know about radicals to begin this exercise by combining the two radicals into one, and... then what?

Please be complete. Thank you!

 

[Math325]

Follow Up

I combined the two radicals and got the root of 5((3^5)*(x^8) all over y^7) i could pull out a 3 and an x but wouldn't know what to do after that?

 

[tvanhooz]

3 * (x/y) * 5th-rt(x^3/y^2)

 

[stapel]

\(\displaystyle \mbox{Simplify }\, \sqrt[5]{\strut \dfrac{27x^2}{y^3}\,}\, \cdot\, \sqrt[5]{\strut \dfrac{9x^6}{y^4}\,}\)

I combined the two radicals and got the root of 5((3^5)*(x^8) all over y^7)

Do you perhaps mean "the fifth root of"...? Because there is no way to convert the original fifth-roots to get a square root, and there is no way to get a five inside that root.

(To learn how to type math as text, please review this article.)

In other words, by "the root of 5((3^5)*(x^8) all over y^7)", did you actually mean the following:

. . . . .5th-rt[ (3^5 x^8) / y^7 ]

...which typesets as:

. . . . .\(\displaystyle \sqrt[5]{\strut \dfrac{3^5\, x^8}{y^7}\,}\)

Kindly please reply with confirmation or else corrections.

i could pull out a 3 and an x but wouldn't know what to do after that?

Assuming I guessed correctly what you meant for the radical, I will guess that, by the above, you mean the following:

. . . . .\(\displaystyle \sqrt[5]{\strut \dfrac{3^5\, x^5\, x^3}{y^5\, y^2}\,}\,=\, \sqrt[5]{\strut 3^5\, x^5\,}\, \sqrt[5]{\strut \dfrac{x^3}{y^5\, y^2}\,}\,=\, (3)\, (x)\, \sqrt[5]{\strut \dfrac{x^3}{y^5\, y^2}\,}\)

If so, you're correct, as far has you've gone. But why did you stop there?