least common multiple 14x^2v^4 and 8u^6x^8v^5

To find the least common multiple (LCM) of two algebraic expressions, we need to consider both the numerical coefficients and the variables with their exponents. 

### Step-by-Step Solution:

1. **Identify the coefficients:**
   - The coefficients are 14 and 8.

2. **Find the LCM of the coefficients:**
   - The prime factorization of 14 is \(2 \times 7\).
   - The prime factorization of 8 is \(2^3\).
   - The LCM of the numerical coefficients is found by taking the highest power of each prime number present in any of the numbers:
     \[
     \text{LCM}(14, 8) = 2^3 \times 7 = 56
     \]

3. **Identify the variables and their exponents:**
   - For \(14x^2v^4\), the variables are \(x^2\) and \(v^4\).
   - For \(8u^6x^8v^5\), the variables are \(u^6\), \(x^8\), and \(v^5\).

4. **Determine the LCM for each variable:**
   - **For \(x\):** The highest power of \(x\) is \(x^8\).
   - **For \(v\):** The highest power of \(v\) is \(v^5\).
   - **For \(u\):** The highest power of \(u\) is \(u^6\) (since the first expression does not contain \(u\), we take the power from the second expression).

5. **Combine the LCM of the coefficients and variables:**
   \[
   \text{LCM}(14x^2v^4, 8u^6x^8v^5) = 56u^6x^8v^5
   \]

### Final Answer:

The least common multiple (LCM) of \(14x^2v^4\) and \(8u^6x^8v^5\) is:
\[
\boxed{56u^6x^8v^5}
\]

Would you like more details on how this was calculated, or do you have any questions?

Here are 5 related questions you might find helpful:
1. How do you find the greatest common divisor (GCD) of algebraic expressions?
2. What is the LCM of two numbers without variables?
3. How do you handle LCM when there are more than two algebraic expressions?
4. Can the LCM of algebraic expressions include negative coefficients?
5. How does the LCM change if one of the terms has a fractional exponent?

**Tip:** When calculating LCMs for algebraic expressions, always consider both the coefficients and the variables separately to ensure accuracy.

Learn how to find the least common multiple (LCM) of algebraic expressions 14x^2v^4 and 8u^6x^8v^5. This problem covers algebraic concepts, including polynomials, exponents, and LCM calculations for algebraic terms. Suitable for students in Grades 9-12.

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