Find the LCM of `14a^2b^3c^4, 20ab^3c^3` and `a^5b^4`.

Answer

To find the LCM (Least Common Multiple) of the expressions \(14a^2b^3c^4\), \(20ab^3c^3\), and \(a^5b^4\), we will follow these steps:

### Step 1: Factor each expression into its prime factors and variables.
1. **For \(14a^2b^3c^4\)**:
- \(14 = 2^1 \times 7^1\)
- So, \(14a^2b^3c^4 = 2^1 \times 7^1 \times a^2 \times b^3 \times c^4\)

2. **For \(20ab^3c^3\)**:
- \(20 = 2^2 \times 5^1\)
- So, \(20ab^3c^3 = 2^2 \times 5^1 \times a^1 \times b^3 \times c^3\)

3. **For \(a^5b^4\)**:
- This is already in its simplest form.
- So, \(a^5b^4 = a^5 \times b^4\)

### Step 2: Identify the highest power of each factor.
- **For \(2\)**:
- Highest power is \(2^2\) (from \(20ab^3c^3\))

- **For \(5\)**:
- Highest power is \(5^1\) (from \(20ab^3c^3\))

- **For \(7\)**:
- Highest power is \(7^1\) (from \(14a^2b^3c^4\))

- **For \(a\)**:
- Highest power is \(a^5\) (from \(a^5b^4\))

- **For \(b\)**:
- Highest power is \(b^4\) (from \(a^5b^4\))

- **For \(c\)**:
- Highest power is \(c^4\) (from \(14a^2b^3c^4\))

### Step 3: Write the LCM using the highest powers identified.
The LCM is given by multiplying all the highest powers together:
\[ \text{LCM} = 2^2 \times 5^1 \times 7^1 \times a^5 \times b^4 \times c^4 \]

### Step 4: Calculate the numerical part.
- \(2^2 = 4\)
- \(5^1 = 5\)
- \(7^1 = 7\)

Now, calculate:
\[ 4 \times 5 = 20 \]
\[ 20 \times 7 = 140 \]

### Step 5: Combine everything to get the final LCM.
Thus, the LCM is:
\[ \text{LCM} = 140a^5b^4c^4 \]

### Final Answer:
\[ \text{LCM} = 140a^5b^4c^4 \]