To simplify
[tex]\sqrt[4]{\dfrac{24x^6y}{128x^4y^5}}[/tex]
we need to use the fact that
[tex]\sqrt[4]{x^4}=|x|[/tex]
Why the absolute value? It's because [tex](-x)^4=(-1)^4x^4=x^4[/tex].
We start by rewriting as
[tex]\sqrt[4]{\dfrac{2^23x^6y}{2^6x^4y^5}}[/tex]
[tex]\sqrt[4]{\dfrac{2^23x^4x^2y}{2^42^2x^4y^4y}}[/tex]
Since [tex]x\neq0[/tex], we have [tex]\dfrac xx=1[/tex], and the above reduces to
[tex]\sqrt[4]{\dfrac{3x^2y}{2^4y^4y}}[/tex]
Then we pull out any 4th powers under the radical, and simplify everything we can:
[tex]\dfrac1{\sqrt[4]{2^4y^4}}\sqrt[4]{\dfrac{3x^2y}{y}}[/tex]
[tex]\dfrac1{|2y|}\sqrt[4]{3x^2}[/tex]
where [tex]y>0[/tex] allows us to write [tex]\dfrac yy=1[/tex], and this also means that [tex]|y|=y[/tex]. So we end up with
[tex]\dfrac{\sqrt[4]{3x^2}}{2y}[/tex]
making the last option the correct answer.
