what is the simplified form of the following expression? assume x≠0. (sqrt5{\frac{10x}{3x^{3}}})…

Explanation:

Step1: Simplify the fraction inside the radical

First, simplify $\frac{10x}{3x^{3}}=\frac{10}{3x^{2}}$ by canceling out one factor of $x$ (since $x\neq0$). So the original expression becomes $\sqrt[5]{\frac{10}{3x^{2}}}$.

Step2: Rationalize the denominator

To rationalize the denominator of the fraction inside the fifth - root, we multiply both the numerator and denominator by $x^{3}$ to make the exponent of $x$ in the denominator a multiple of 5. We get $\sqrt[5]{\frac{10x^{3}}{3x^{5}}}$.

Step3: Separate the radical

We can write $\sqrt[5]{\frac{10x^{3}}{3x^{5}}}=\frac{\sqrt[5]{10x^{3}}}{\sqrt[5]{3x^{5}}}$. Since $\sqrt[5]{3x^{5}} = x\sqrt[5]{3}$, the expression is $\frac{\sqrt[5]{10x^{3}}}{x\sqrt[5]{3}}$.

Step4: Multiply by a form of 1 to get a single radical in the numerator

Multiply the fraction by $\frac{\sqrt[5]{3^{4}}}{\sqrt[5]{3^{4}}}$ (which is 1). We have $\frac{\sqrt[5]{10x^{3}}\times\sqrt[5]{3^{4}}}{x\sqrt[5]{3}\times\sqrt[5]{3^{4}}}=\frac{\sqrt[5]{10x^{3}\times81}}{x\sqrt[5]{3^{5}}}=\frac{\sqrt[5]{810x^{3}}}{3x}$.

Answer:

$\frac{\sqrt[5]{810x^{3}}}{3x}$