To simplify the expression $(5ab^3)^2(-3a^4)$, follow these steps:

1. Expand the first term: $(5ab^3)^2 = (5^2)(a^2)(b^{3 \cdot 2}) = 25a^2b^6$

2. Multiply with $-3a^4$: Combine the like terms:

   • Coefficients: $25 \times -3 = -75$
   • $a$-terms: $a^2 \cdot a^4 = a^{2+4} = a^6$
   • $b$-terms remain $b^6$ as there are no other $b$-terms.

3. Final result: $-75a^6b^6$

Simplified Expression:

$-75a^6b^6$

Would you like me to elaborate on any specific part of the process?

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Related Questions:

1. How do you handle negative exponents in similar expressions?
2. Can you explain the rules for multiplying exponents in more depth?
3. What are the general rules for expanding powers with multiple variables?
4. How do you distribute powers across parentheses containing sums?
5. Could you verify this using numerical substitution for $a$ and $b$?

Tip:

Always simplify exponents step by step to avoid mistakes, especially when working with multiple variables and coefficients.