The problem is to simplify the expression $\frac{18a^2b^3}{10a^3b^4} \times \frac{9a^3b^4}{3a^4b^8}$.

1. Problem Description

The problem is to simplify the expression $\frac{18a^2b^3}{10a^3b^4} \times \frac{9a^3b^4}{3a^4b^8}$.

2. Solution Steps

First, we can multiply the numerators and denominators:

$\frac{18a^2b^3}{10a^3b^4} \times \frac{9a^3b^4}{3a^4b^8} = \frac{18 \times 9 \times a^2 \times a^3 \times b^3 \times b^4}{10 \times 3 \times a^3 \times a^4 \times b^4 \times b^8}$

Now, we simplify the constants:

$18 \times 9 = 162$

$10 \times 3 = 30$

So, the expression becomes:

$\frac{162 a^2 a^3 b^3 b^4}{30 a^3 a^4 b^4 b^8}$

Now, we simplify the variables using the exponent rules $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$:

$a^2 \times a^3 = a^{2+3} = a^5$

$b^3 \times b^4 = b^{3+4} = b^7$

$a^3 \times a^4 = a^{3+4} = a^7$

$b^4 \times b^8 = b^{4+8} = b^{12}$

So, the expression becomes:

$\frac{162 a^5 b^7}{30 a^7 b^{12}}$

Now, we simplify the constants $\frac{162}{30}$ by dividing both numerator and denominator by 6:

$\frac{162}{6} = 27$

$\frac{30}{6} = 5$

So, the expression becomes:

$\frac{27 a^5 b^7}{5 a^7 b^{12}}$

Now, we simplify the variables:

$\frac{a^5}{a^7} = a^{5-7} = a^{-2} = \frac{1}{a^2}$

$\frac{b^7}{b^{12}} = b^{7-12} = b^{-5} = \frac{1}{b^5}$

Therefore, the expression becomes:

$\frac{27}{5} \times \frac{1}{a^2} \times \frac{1}{b^5} = \frac{27}{5a^2b^5}$

3. Final Answer

$\frac{27}{5a^2b^5}$