which expression is equivalent to $sqrt4{\frac{24x^{6}y}{128x^{4}y^{5}}}$? assume $x\neq0$ and $y >…

which expression is equivalent to $sqrt4{\frac{24x^{6}y}{128x^{4}y^{5}}}$? assume $x\neq0$ and $y > 0$.\n$\frac{sqrt4{3}}{2x^{2}y}$\n$\frac{x(sqrt4{3})}{4y^{2}}$\n$\frac{sqrt4{3}}{4xy^{2}}$\n$\frac{sqrt4{3x^{2}}}{2y}$

which expression is equivalent to $sqrt4{\frac{24x^{6}y}{128x^{4}y^{5}}}$? assume $x\neq0$ and $y > 0$.\n$\frac{sqrt4{3}}{2x^{2}y}$\n$\frac{x(sqrt4{3})}{4y^{2}}$\n$\frac{sqrt4{3}}{4xy^{2}}$\n$\frac{sqrt4{3x^{2}}}{2y}$

Answer

Explanation:

Step1: Simplify the fraction inside the fourth - root

Simplify $\frac{24x^{6}y}{128x^{4}y^{5}}$ by dividing the coefficients and using the rule of exponents $\frac{a^{m}}{a^{n}}=a^{m - n}$. The coefficient $\frac{24}{128}=\frac{3}{16}$, and for the $x$ - terms $\frac{x^{6}}{x^{4}}=x^{6 - 4}=x^{2}$, for the $y$ - terms $\frac{y}{y^{5}}=y^{1-5}=y^{-4}$. So, $\frac{24x^{6}y}{128x^{4}y^{5}}=\frac{3x^{2}}{16y^{4}}$.

Step2: Apply the fourth - root

$\sqrt[4]{\frac{3x^{2}}{16y^{4}}}=\frac{\sqrt[4]{3x^{2}}}{\sqrt[4]{16y^{4}}}$. Since $\sqrt[4]{16y^{4}} = 2y$, the expression becomes $\frac{\sqrt[4]{3x^{2}}}{2y}$.

Answer:

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