Find the LCM of
`6x ^(2) y ^(3), 8x ^(3) y ^(2) , 12 x ^(4) y ^(3) and 10 x ^(3) y ^(4).`

Options

A. `120 x ^(4) y ^(4)`
B. `240 x ^(5) y ^(4)`
C. `60 x ^(4) y ^(5)`
D. `120 x ^(5) y ^(5)`

Answer

To find the LCM (Least Common Multiple) of the expressions \(6x^2y^3\), \(8x^3y^2\), \(12x^4y^3\), and \(10x^3y^4\), we will follow these steps:

### Step 1: Factor each term
First, we need to factor each coefficient and identify the powers of \(x\) and \(y\) in each term.

1. **For \(6x^2y^3\)**:
- Coefficient: \(6 = 2 \times 3\)
- Powers: \(x^2\), \(y^3\)

2. **For \(8x^3y^2\)**:
- Coefficient: \(8 = 2^3\)
- Powers: \(x^3\), \(y^2\)

3. **For \(12x^4y^3\)**:
- Coefficient: \(12 = 2^2 \times 3\)
- Powers: \(x^4\), \(y^3\)

4. **For \(10x^3y^4\)**:
- Coefficient: \(10 = 2 \times 5\)
- Powers: \(x^3\), \(y^4\)

### Step 2: Determine the LCM of the coefficients
Next, we find the LCM of the coefficients \(6\), \(8\), \(12\), and \(10\):

- **Prime factorization**:
- \(6 = 2^1 \times 3^1\)
- \(8 = 2^3\)
- \(12 = 2^2 \times 3^1\)
- \(10 = 2^1 \times 5^1\)

- **Take the highest power of each prime**:
- For \(2\): Highest power is \(2^3\) (from \(8\))
- For \(3\): Highest power is \(3^1\) (from \(6\) or \(12\))
- For \(5\): Highest power is \(5^1\) (from \(10\))

Thus, the LCM of the coefficients is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \]

### Step 3: Determine the LCM of the variables
Now we find the LCM of the variable parts:

- For \(x\): The highest power is \(x^4\) (from \(12x^4y^3\)).
- For \(y\): The highest power is \(y^4\) (from \(10x^3y^4\)).

### Step 4: Combine the results
Now we combine the LCM of the coefficients and the LCM of the variables:
\[ LCM = 120x^4y^4 \]

### Final Answer
Thus, the LCM of \(6x^2y^3\), \(8x^3y^2\), \(12x^4y^3\), and \(10x^3y^4\) is:
\[ \boxed{120x^4y^4} \]

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