To find the least common multiple (LCM) of two expressions, we follow these steps:

1. Factor the constants (numerical part):

   • $10 = 2 \times 5$
   • $4 = 2^2$

2. Find the highest powers of each prime factor in the constants:

   • The highest power of 2 is $2^2$ (from 4).
   • The highest power of 5 is $5$ (from 10).

3. Compare the variable part:

   • $a^2$ and $a^3$, the highest power of $a$ is $a^3$.

4. Multiply the highest powers together: $\text{LCM}(10a^2, 4a^3) = 2^2 \times 5 \times a^3 = 4 \times 5 \times a^3 = 20a^3$

So, the least common multiple (LCM) of $10a^2$ and $4a^3$ is $\mathbf{20a^3}$.

Would you like any more details on this? Here are some related questions:

1. How do you find the greatest common divisor (GCD) of two algebraic expressions?
2. What is the process of factoring algebraic expressions?
3. How does the LCM of polynomials differ from that of numbers?
4. What is the connection between LCM and GCD in algebra?
5. How does exponent comparison help in finding LCM in algebraic expressions?

Tip: Always factor both the constants and variables separately when finding the LCM of algebraic expressions.