To solve the given problem, we need to simplify the expression $\frac{\sqrt{8x^{10}}}{\sqrt{8x^3}}$.

First, use the property of radicals: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$.

$\frac{\sqrt{8x^{10}}}{\sqrt{8x^3}} = \sqrt{\frac{8x^{10}}{8x^3}}$

Simplify the expression inside the square root:

$\frac{8x^{10}}{8x^3} = x^{10-3} = x^7$

Thus, the expression becomes:

$\sqrt{x^7}$

Express $\sqrt{x^7}$ in simplest radical form:

$\sqrt{x^7} = \sqrt{x^6 \cdot x} = \sqrt{(x^3)^2 \cdot x} = x^3\sqrt{x}$

So, the simplified expression is:

$x^3\sqrt{x}$