https://file.aiquickdraw.com/mathbot/file/17284335987923byoemy7.jpg

The expression in the image is:

\[
\frac{\frac{3}{x} - 8x^2 + 16x}{\frac{3}{x^3} - 4x^2}
\]

### Steps to simplify:

1. **Simplify the numerator**:  
   The numerator is:
   \[
   \frac{3}{x} - 8x^2 + 16x
   \]
   To combine terms, we need to find a common denominator. The least common denominator (LCD) between \(x\) and the other terms is \(x\). So:
   \[
   \frac{3}{x} = \frac{3}{x}, \quad -8x^2 = \frac{-8x^3}{x}, \quad 16x = \frac{16x^2}{x}
   \]
   Now the numerator becomes:
   \[
   \frac{3 - 8x^3 + 16x^2}{x}
   \]

2. **Simplify the denominator**:  
   The denominator is:
   \[
   \frac{3}{x^3} - 4x^2
   \]
   Again, we need a common denominator. The LCD here is \(x^3\), so:
   \[
   \frac{3}{x^3} = \frac{3}{x^3}, \quad -4x^2 = \frac{-4x^5}{x^3}
   \]
   The denominator becomes:
   \[
   \frac{3 - 4x^5}{x^3}
   \]

3. **Combine the two fractions**:  
   We can now rewrite the entire expression as:
   \[
   \frac{\frac{3 - 8x^3 + 16x^2}{x}}{\frac{3 - 4x^5}{x^3}}
   \]
   Dividing one fraction by another is the same as multiplying by the reciprocal. So:
   \[
   \frac{3 - 8x^3 + 16x^2}{x} \times \frac{x^3}{3 - 4x^5}
   \]

4. **Simplify the resulting expression**:  
   The \(x\) in the numerator cancels with one of the \(x^3\) terms in the denominator, leaving us with:
   \[
   \frac{(3 - 8x^3 + 16x^2) x^2}{3 - 4x^5}
   \]

There is no further simplification possible without specific factorization techniques. The final simplified expression is:

\[
\frac{(3 - 8x^3 + 16x^2) x^2}{3 - 4x^5}
\]

Would you like further details on any step, or have any specific questions? 

### Related Questions:
1. How can I simplify a complex rational expression with multiple terms in the numerator and denominator?
2. What are the steps to finding a common denominator when simplifying fractions with variables?
3. How do you cancel terms when simplifying complex rational expressions?
4. How do you factor higher-degree polynomials like \(3 - 8x^3 + 16x^2\)?
5. What are some common techniques for simplifying algebraic fractions?

**Tip:** Always remember to check if any common factors can be canceled out before multiplying or dividing expressions.

Simplify the expression (3/x - 8x^2 + 16x) / (3/x^3 - 4x^2)

Simplify the complex rational expression (3/x - 8x^2 + 16x) / (3/x^3 - 4x^2) with step-by-step guidance. This problem covers algebraic concepts like finding a common denominator, simplifying rational expressions, and properties of exponents. Suitable for high school students in Grades 10-12.

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