which expression is equivalent to $sqrt3{\frac{10x^{5}}{54x^{8}}}$? assume $x\neq0$.\n$\frac{sqrt3{10x}}{3x^{…

which expression is equivalent to $sqrt3{\frac{10x^{5}}{54x^{8}}}$? assume $x\neq0$.\n$\frac{sqrt3{10x}}{3x^{2}}$\n$\frac{x(sqrt3{5x})}{3}$\n$\frac{3(sqrt3{5x})}{x}$\n$\frac{sqrt3{5}}{3x}$

which expression is equivalent to $sqrt3{\frac{10x^{5}}{54x^{8}}}$? assume $x\neq0$.\n$\frac{sqrt3{10x}}{3x^{2}}$\n$\frac{x(sqrt3{5x})}{3}$\n$\frac{3(sqrt3{5x})}{x}$\n$\frac{sqrt3{5}}{3x}$

Answer

Explanation:

Step1: Simplify the fraction inside the cube - root

Use the quotient - rule of exponents $\frac{a^m}{a^n}=a^{m - n}$ and simplify the coefficient. $\frac{10x^{5}}{54x^{8}}=\frac{10}{54}\times x^{5-8}=\frac{5}{27}x^{- 3}$. So, $\sqrt[3]{\frac{10x^{5}}{54x^{8}}}=\sqrt[3]{\frac{5}{27}x^{-3}}$.

Step2: Apply the cube - root to each part

Use the property $\sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b}$. $\sqrt[3]{\frac{5}{27}x^{-3}}=\sqrt[3]{\frac{5}{27}}\cdot\sqrt[3]{x^{-3}}$. Since $\sqrt[3]{\frac{5}{27}}=\frac{\sqrt[3]{5}}{\sqrt[3]{27}}=\frac{\sqrt[3]{5}}{3}$ and $\sqrt[3]{x^{-3}} = x^{-1}=\frac{1}{x}$, then $\sqrt[3]{\frac{5}{27}x^{-3}}=\frac{\sqrt[3]{5}}{3x}$.

Answer:

$\frac{\sqrt[3]{5}}{3x}$