Let's simplify the given expression step-by-step:
[tex]\[
\sqrt[5]{\frac{10 x}{3 x^3}}
\][/tex]
1. Simplify the expression inside the 5th root:
Observe that the original expression inside the fifth root is a fraction:
[tex]\[
\frac{10 x}{3 x^3}
\][/tex]
2. Simplify the fraction:
We can simplify the fraction by combining like terms:
[tex]\[
\frac{10 x}{3 x^3} = \frac{10}{3} \cdot \frac{x}{x^3}
\][/tex]
Recall that we can further simplify \(\frac{x}{x^3}\) using the properties of exponents:
[tex]\[
\frac{x}{x^3} = x^{1-3} = x^{-2}
\][/tex]
Thus, the simplified fraction becomes:
[tex]\[
\frac{10 x}{3 x^3} = \frac{10}{3} \cdot x^{-2} = \frac{10}{3 x^2}
\][/tex]
3. Apply the 5th root to the simplified fraction:
Now, we take the fifth root of the simplified expression:
[tex]\[
\sqrt[5]{\frac{10}{3 x^2}}
\][/tex]
4. Simplify further by splitting the fraction under the 5th root:
We can apply the fifth root to both the numerator and the denominator separately:
[tex]\[
\sqrt[5]{\frac{10}{3 x^2}} = \frac{\sqrt[5]{10}}{\sqrt[5]{3 x^2}}
\][/tex]
5. Simplify the denominator by applying the 5th root:
Split the fifth root further in the denominator:
[tex]\[
\sqrt[5]{3 x^2} = \sqrt[5]{3} \cdot \sqrt[5]{x^2}
\][/tex]
Using the properties of exponents, we know:
[tex]\[
\sqrt[5]{x^2} = x^{2/5}
\][/tex]
So, the denominator becomes:
[tex]\[
\sqrt[5]{3 x^2} = 3^{1/5} \cdot x^{2/5}
\][/tex]
6. Final simplified form:
Now, place the simplified numerator and denominator together:
[tex]\[
\frac{\sqrt[5]{10}}{3^{1/5} \cdot x^{2/5}}
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
\frac{\sqrt[5]{10}}{3^{1/5} \cdot x^{2/5}}
\][/tex]
