We are asked to simplify the expression:

$\left(\frac{-1}{5}k^5 m^3\right)^3$

Step-by-Step Simplification:

1. Apply the power to each part of the expression: $\left(\frac{-1}{5}k^5 m^3\right)^3 = \frac{(-1)^3}{5^3} (k^5)^3 (m^3)^3$

2. Simplify each part:

   • $(-1)^3 = -1$
   • $5^3 = 125$
   • $(k^5)^3 = k^{15}$ (multiply the exponents)
   • $(m^3)^3 = m^9$

3. Put it all together: $\frac{-1}{125} k^{15} m^9$

Thus, the simplified expression is:

$\frac{-1}{125} k^{15} m^9$

Would you like further explanation or have any questions about this solution?

Here are 5 related questions:

1. How do you simplify an expression with multiple variables raised to powers?
2. What is the rule for multiplying exponents when raising a power to a power?
3. Can you explain how to handle negative bases in exponentiation?
4. How do you apply exponent rules to fractions?
5. How do you simplify expressions involving negative exponents?

Tip: When raising a fraction to a power, remember to apply the exponent to both the numerator and denominator.