To simplify the expression \(\left(\frac{2x^3}{5x^4}\right)^{-3}\), follow these steps:

1. **Simplify the expression inside the parentheses**:
   - The coefficients: \(\frac{2}{5}\) cannot be simplified further, so it stays as \(\frac{2}{5}\).
   - The exponents of \(x\): Subtract the exponents \(3 - 4 = -1\), so the expression inside the parentheses simplifies to \(\frac{2}{5x}\).

   Therefore:
   \[
   \frac{2x^3}{5x^4} = \frac{2}{5x}
   \]

2. **Apply the negative exponent**: Raise both the numerator and the denominator to the power of \(-3\).

   \[
   \left(\frac{2}{5x}\right)^{-3} = \left(\frac{2^{-3}}{(5x)^{-3}}\right)
   \]

   This is equivalent to taking the reciprocal of the expression and raising everything to the positive exponent \(3\):
   \[
   = \left(\frac{(5x)^3}{2^3}\right)
   \]

3. **Simplify the expression**:
   - \(2^3 = 8\)
   - \((5x)^3 = 5^3 \times x^3 = 125x^3\)

   Therefore:
   \[
   \left(\frac{2x^3}{5x^4}\right)^{-3} = \frac{125x^3}{8}
   \]

So, \(\left(\frac{2x^3}{5x^4}\right)^{-3} = \frac{125x^3}{8}\).

Would you like more details or have any questions about this process?

Here are 5 related questions you might find interesting:
1. How would you simplify \(\left(\frac{3x^2}{4y^3}\right)^{-2}\)?
2. What is the result of \(\left(\frac{5x^5}{2x^2}\right)^{-2}\)?
3. How do you simplify \(\left(\frac{7y^3}{2x^4}\right)^2\)?
4. Can you solve \(\left(\frac{3a^4}{9b^2}\right)^{-1}\)?
5. How do you simplify \(\left(\frac{x^2 + 3x}{x^3}\right)^{-2}\)?

**Tip:** When dealing with negative exponents, remember that they indicate the reciprocal of the base raised to the positive exponent.

Learn how to simplify the expression (2x^3)/(5x^4)^-3 with a detailed step-by-step solution. This problem involves applying rules of exponents, negative exponents, and fractional exponents. Suitable for high school students.

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