Question

The least common multiple of 54c 2de 3 and 38c 4 is _____.

342 c 2
342 c 4de 3
1,026 c 4de 3
1,026 c 4

Answer 1

To find the least common multiple (LCM) of two polynomials, we'll first break down the individual polynomials to their factors.

1. **Factor each polynomial:**
- \(54c^2de^3\): We break this down into numerical and variable parts:
- The numerical part \(54 = 2 \times 3^3\) (since \(54 = 2 \cdot 27 = 2 \cdot 3 \cdot 9 = 2 \cdot 3^3\)).
- The variable part is \(c^2d^1e^3\).

Thus, we can write:
\[
54c^2de^3 = 2 \cdot 3^3 \cdot c^2 \cdot d^1 \cdot e^3
\]

- \(38c^4\): Similarly, we break this down:
- The numerical part \(38 = 2 \cdot 19\).
- The variable part is \(c^4\).

Thus:
\[
38c^4 = 2 \cdot 19 \cdot c^4
\]

2. **Calculate the LCM:**
- For the numerical part, we take the highest powers of each prime factor:
- For \(2\), the highest power is \(2^1\) (appears in both).
- For \(3\), the highest power is \(3^3\) (from \(54\)).
- For \(19\), the highest power is \(19^1\) (from \(38\)).

So for the coefficients, we get:
\[
LCM(54, 38) = 2^1 \cdot 3^3 \cdot 19^1 = 2 \cdot 27 \cdot 19 = 2 \cdot 513 = 1026
\]

- For the variable parts, we take the highest powers:
- For \(c\), we take \(c^4\) (the highest power from \(38c^4\)).
- For \(d\), we take \(d^1\) (appearing in \(54c^2de^3\)).
- For \(e\), we take \(e^3\) (from \(54c^2de^3\)).

Thus, we get:
\[
c^4d^1e^3
\]

3. **Combine the results:**
- The LCM is:
\[
LCM(54c^2de^3, 38c^4) = 1026c^4de^3
\]

From the provided options, the equivalent form of our result is \(1,026c^4de^3\).

Thus, the answer is:
\[
\boxed{1,026c^4de^3}
\]