The LCM of `12x^(2)y^(3)z^(2)` and `18x^(4)y^(2)z^(3)` is

Options

A. `24x^(4)y^(2)z^(2)`
B. `32x^(4)yz^(3)`
C. `36x^(4)y^(3)z^(3)`
D. 21xyz

Answer

To find the LCM of the expressions \(12x^{2}y^{3}z^{2}\) and \(18x^{4}y^{2}z^{3}\), we will follow these steps:

### Step 1: Factor the coefficients
First, we need to factor the coefficients of the two expressions.

- For \(12\):
\[ 12 = 2^2 \times 3^1 \]

- For \(18\):
\[ 18 = 2^1 \times 3^2 \]

### Step 2: Identify the variables and their powers
Next, we identify the variables and their respective powers in both expressions.

- For \(12x^{2}y^{3}z^{2}\):
- \(x\) has a power of \(2\)
- \(y\) has a power of \(3\)
- \(z\) has a power of \(2\)

- For \(18x^{4}y^{2}z^{3}\):
- \(x\) has a power of \(4\)
- \(y\) has a power of \(2\)
- \(z\) has a power of \(3\)

### Step 3: Determine the LCM of the coefficients
To find the LCM of the coefficients, we take the highest power of each prime factor:

- For \(2\):
- The maximum power is \(2^2\) (from \(12\))

- For \(3\):
- The maximum power is \(3^2\) (from \(18\))

Thus, the LCM of the coefficients is:
\[ LCM(12, 18) = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 4: Determine the LCM of the variables
To find the LCM of the variables, we take the highest power of each variable:

- For \(x\):
- The maximum power is \(x^4\) (from \(18x^{4}y^{2}z^{3}\))

- For \(y\):
- The maximum power is \(y^3\) (from \(12x^{2}y^{3}z^{2}\))

- For \(z\):
- The maximum power is \(z^3\) (from \(18x^{4}y^{2}z^{3}\))

### Step 5: Combine the LCM of coefficients and variables
Now we combine the LCM of the coefficients and the variables:
\[ LCM(12x^{2}y^{3}z^{2}, 18x^{4}y^{2}z^{3}) = 36x^{4}y^{3}z^{3} \]

### Final Answer
Thus, the LCM of \(12x^{2}y^{3}z^{2}\) and \(18x^{4}y^{2}z^{3}\) is:
\[ \boxed{36x^{4}y^{3}z^{3}} \]