Let's simplify the given expression step-by-step:

Expression:

$\frac{18a^3b^{-2} \cdot 5ab^2}{9a^4b \cdot (-ab^3)}$

Step 1: Simplify the numerator and the denominator separately.

Numerator:

$18a^3b^{-2} \cdot 5ab^2 = (18 \cdot 5) \cdot (a^3 \cdot a) \cdot (b^{-2} \cdot b^2)$

$= 90a^4b^0$

Since $b^0 = 1$, the simplified numerator is:

$90a^4$

Denominator:

$9a^4b \cdot (-ab^3) = (9 \cdot -1) \cdot (a^4 \cdot a) \cdot (b \cdot b^3)$

$= -9a^5b^4$

Step 2: Combine the simplified numerator and denominator.

$\frac{90a^4}{-9a^5b^4}$

Step 3: Simplify the fraction.

Simplify the coefficients:

$\frac{90}{-9} = -10$

Simplify the powers of $a$:

$\frac{a^4}{a^5} = \frac{1}{a} = a^{-1}$

Simplify the powers of $b$:

$\frac{1}{b^4} = b^{-4}$

Final Simplified Expression:

$-10a^{-1}b^{-4} = \frac{-10}{ab^4}$

Answer:

$\frac{-10}{ab^4}$

Would you like further details or have any questions?

Related Questions:

1. How do you simplify expressions with negative exponents?
2. What are the rules for multiplying and dividing powers of variables?
3. How can coefficients be simplified in fraction form?
4. How do negative signs affect the simplification process?
5. How can we handle terms with zero exponents?

Tip: When simplifying expressions with variables, handle each variable and exponent separately to ensure accuracy!