Find the least common multiple. $$ 5 j^{4}, 6 j^{5} $$ Write your answer as a constant times a product of single variables raised to exponents.

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To find the least common multiple (LCM) of the expressions $$ 5 j^{4} $$ and $$ 6 j^{5} $$, we will follow these steps:

1. **Identify the coefficients**: The coefficients are $$ 5 $$ and $$ 6 $$.
2. **Find the LCM of the coefficients**: The LCM of $$ 5 $$ and $$ 6 $$ can be calculated by finding the smallest multiple that both numbers share.
3. **Identify the variable parts**: The variable parts are $$ j^{4} $$ and $$ j^{5} $$.
4. **Find the LCM of the variable parts**: The LCM of $$ j^{4} $$ and $$ j^{5} $$ is determined by taking the variable with the highest exponent.

Now, let's calculate the LCM step by step.

### Step 1: LCM of the coefficients
The prime factorization of the coefficients:
- $$ 5 = 5^1 $$
- $$ 6 = 2^1 \times 3^1 $$

The LCM is found by taking the highest power of each prime:
- For $$ 2 $$: $$ 2^1 $$
- For $$ 3 $$: $$ 3^1 $$
- For $$ 5 $$: $$ 5^1 $$

Thus, the LCM of the coefficients is:
$$
\text{LCM}(5, 6) = 2^1 \times 3^1 \times 5^1 = 30
$$

### Step 2: LCM of the variable parts
The variable parts are $$ j^{4} $$ and $$ j^{5} $$. The LCM is:
$$
\text{LCM}(j^{4}, j^{5}) = j^{\max(4, 5)} = j^{5}
$$

### Final LCM
Combining both results, we have:
$$
\text{LCM}(5 j^{4}, 6 j^{5}) = 30 j^{5}
$$

Thus, the least common multiple is:
$$
\boxed{30 j^{5}}
$$
The least common multiple is $$ 30 j^{5} $$.

Find the least common multiple. \( 5 j^{4}, 6 j^{5} \) Write your answer as a constant times a product of single variables raised to exponents.

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