The given equation is:

$(f - g)(x^3) = \frac{(x^3)^2}{x^3 + 3} - \frac{\sqrt{x^3}}{x^3}$

To simplify this equation, let's break it down step by step.

Step 1: Simplify $(x^3)^2$

$(x^3)^2 = x^6$

Step 2: Substitute into the equation

Now we substitute $x^6$ back into the equation: $(f - g)(x^3) = \frac{x^6}{x^3 + 3} - \frac{\sqrt{x^3}}{x^3}$

Step 3: Simplify the second term $\frac{\sqrt{x^3}}{x^3}$

$\frac{\sqrt{x^3}}{x^3} = \frac{x^{3/2}}{x^3} = x^{3/2 - 3} = x^{-3/2}$

Final Equation

The equation now becomes: $(f - g)(x^3) = \frac{x^6}{x^3 + 3} - x^{-3/2}$

This is the simplified form of the given expression.

Would you like to explore further details or solve this for specific $f(x)$ and $g(x)$?

Here are 5 related questions to expand on this topic:

1. How do you simplify expressions with higher exponents like $x^6$?
2. What are the properties of exponents that help simplify $x^{a} / x^{b}$?
3. How do you handle square roots of variables like $\sqrt{x^3}$?
4. Can you solve for $f(x)$ and $g(x)$ given other constraints?
5. How does the behavior of the function change as $x \to 0$ or $x \to \infty$?

Tip: When dealing with fractional exponents, remember that $x^{m/n} = \sqrt[n]{x^m}$, which is useful for simplifications.