To simplify the expression $\left(\frac{2a^3}{3y^3}\right) \times \left(\frac{6y^3}{16x^2}\right)$, follow these steps:

1. Multiply the numerators: $2a^3 \times 6y^3 = 12a^3y^3$

2. Multiply the denominators: $3y^3 \times 16x^2 = 48y^3x^2$

3. Combine the fractions: $\frac{12a^3y^3}{48y^3x^2}$

4. Simplify the fraction:

   • First, cancel out the $y^3$ from both the numerator and the denominator: $\frac{12a^3\cancel{y^3}}{48\cancel{y^3}x^2} = \frac{12a^3}{48x^2}$
   • Next, simplify the numerical coefficient $\frac{12}{48}$: $\frac{12}{48} = \frac{1}{4}$
   • Therefore, the simplified expression is: $\frac{a^3}{4x^2}$

So, the simplified expression is $\boxed{\frac{a^3}{4x^2}}$.

Would you like more details on any step, or do you have further questions?

Here are some related questions you might find helpful:

1. How do you simplify a fraction involving variables?
2. What are the rules for canceling terms in a fraction?
3. How do you multiply fractions with variables?
4. How can you factor common terms in algebraic expressions?
5. What are some tips for simplifying complex algebraic fractions?

Tip: Always look for common factors in the numerator and denominator to simplify fractions more easily.