Question

Find the least common multiple.
20k8, 45k4
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 numbers \(20k^8\) and \(45k^4\), we need to determine the LCM of the coefficients (20 and 45) and the LCM of the variable terms (\(k^8\) and \(k^4\)).

**Step 1: Find the LCM of the coefficients (20 and 45)**

First, we can find the prime factorization of both coefficients:
- \(20 = 2^2 \times 5^1\)
- \(45 = 3^2 \times 5^1\)

To find the LCM, we take the highest power of each prime factor present in both factorizations:
- The highest power of \(2\) is \(2^2\).
- The highest power of \(3\) is \(3^2\).
- The highest power of \(5\) is \(5^1\).

Thus, the LCM of the coefficients is:
\[
LCM(20, 45) = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180.
\]

**Step 2: Find the LCM of the variable parts (\(k^8\) and \(k^4\))**

The LCM for the variable parts is determined by taking the highest exponent:
- The highest exponent for \(k\) is \(k^8\).

**Step 3: Combine the results**

Putting both parts together, the LCM of \(20k^8\) and \(45k^4\) is:
\[
LCM(20k^8, 45k^4) = 180k^8.
\]

Thus, the least common multiple is:
\[
\boxed{180k^8}.
\]