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: \[ \boxed{180k^8}. \]