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To solve the problem of finding the least common multiple (LCM) of \( 4v^2 \) and \( 5a^4 \), we need to follow these steps:

### Step 1: Analyze the given expressions
- The terms are \( 4v^2 \) and \( 5a^4 \).
- \( 4v^2 \) includes the coefficient 4 and the variable \( v^2 \).
- \( 5a^4 \) includes the coefficient 5 and the variable \( a^4 \).

### Step 2: Find the LCM of the numerical coefficients
- The coefficients are 4 and 5.
- The LCM of 4 and 5 is 20 because 4 and 5 are relatively prime (no common factors).

### Step 3: Address the variable parts
- \( v^2 \) and \( a^4 \) have different variables.
- When finding the LCM of terms with different variables, we simply include both variables in the LCM, because they do not share any common powers.

### Step 4: Combine everything
- The LCM of the numerical coefficients is 20.
- The LCM of the variable part is \( v^2 \) and \( a^4 \), so we include both \( v^2 \) and \( a^4 \).

Thus, the least common multiple (LCM) of \( 4v^2 \) and \( 5a^4 \) is:

\[
\boxed{20v^2a^4}
\]

Let me know if you need further details or have any questions!

### Related Questions:
1. How do you find the LCM of polynomials with multiple variables?
2. What is the general formula for finding the LCM of algebraic expressions?
3. How do you simplify expressions involving least common multiples?
4. Can the LCM of two algebraic expressions ever be smaller than one of the terms?
5. How does the greatest common divisor (GCD) relate to finding the LCM?

#### Tip:
Always factor each term completely when finding the least common multiple of algebraic expressions to ensure no common factors are overlooked!

Find the least common multiple of 4v^2 and 5a^4.

This solution shows how to find the least common multiple of the algebraic expressions 4v^2 and 5a^4. It covers key algebraic concepts, including how to find the LCM of coefficients and variables. Suitable for students in Grades 9-11.

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