Question

Find the least common multiple.
80d2, 40d4
Write your answer as a constant times a product of single variables raised to exponents.

Answer 1

To find the least common multiple (LCM) of the two expressions \(80d^2\) and \(40d^4\), we first break down each expression into its prime factors and variable components.

1. **Factorizing the constants:**
- \(80 = 8 \times 10 = 2^4 \times 2 \times 5 = 2^4 \times 5\)
- \(40 = 4 \times 10 = 2^2 \times 2 \times 5 = 2^3 \times 5\)

2. **Identifying the variable parts:**
- The variable part for the first expression is \(d^2\).
- The variable part for the second expression is \(d^4\).

3. **Finding the LCM of the constants:**
- The LCM of the constants \(80\) and \(40\) can be found by taking the highest power of each prime factor:
- For \(2\), the maximum power is \(2^4\) (from \(80\)).
- For \(5\), the maximum power is \(5^1\) (from both).
- Thus, the LCM of the constants is:
\[
LCM(80, 40) = 2^4 \times 5^1 = 80
\]

4. **Finding the LCM of the variable parts:**
- The LCM of the variable parts \(d^2\) and \(d^4\) is determined by taking the highest power:
- The maximum power of \(d\) is \(d^4\).
- Therefore, the LCM of the variable parts is:
\[
LCM(d^2, d^4) = d^4
\]

5. **Combining the results:**
- The overall LCM is the product of the LCM of the constants and the LCM of the variable parts:
\[
LCM(80d^2, 40d^4) = 80 \times d^4
\]

Thus, the least common multiple \(LCM(80d^2, 40d^4)\) is:

\[
\boxed{80d^4}
\]