Question

I have the question in a screengrab. Thank you, whoever helps me.

Answer 1

To simplify

$\sqrt[4]{\dfrac{24x^6y}{128x^4y^5}}$

we need to use the fact that

$\sqrt[4]{x^4}=|x|$

Why the absolute value? It's because $(-x)^4=(-1)^4x^4=x^4$.

We start by rewriting as

$\sqrt[4]{\dfrac{2^23x^6y}{2^6x^4y^5}}$

$\sqrt[4]{\dfrac{2^23x^4x^2y}{2^42^2x^4y^4y}}$

Since $x\neq0$, we have $\dfrac xx=1$, and the above reduces to

$\sqrt[4]{\dfrac{3x^2y}{2^4y^4y}}$

Then we pull out any 4th powers under the radical, and simplify everything we can:

$\dfrac1{\sqrt[4]{2^4y^4}}\sqrt[4]{\dfrac{3x^2y}{y}}$

$\dfrac1{|2y|}\sqrt[4]{3x^2}$

where $y>0$ allows us to write $\dfrac yy=1$, and this also means that $|y|=y$. So we end up with

$\dfrac{\sqrt[4]{3x^2}}{2y}$

making the last option the correct answer.