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(f - g) (x³) = (x³)² over x³ + 3 minus 

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 \(g(x)\)? 

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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.

(f - g)(x³) = (x³)² / (x³ + 3) - √(x³) / x³

This problem demonstrates how to simplify an expression involving powers of x, square roots, and division. The solution includes breaking down exponents and handling fractional powers, suitable for high school students in Grades 10-12.

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