The LCM of `2^(3) xx 3^(2) and 2^(2) xx 3^(2)` is :

Options

A. `2^(3)`
B. `3^(3)`
C. `2^(3) xx 3^(3)`
D. `2^(2) xx 3^(2)`

Answer

To find the LCM of the numbers \(2^3 \times 3^2\) and \(2^2 \times 3^2\), we can follow these steps:

### Step 1: Identify the prime factorization of both numbers
- The first number \(A = 2^3 \times 3^2\)
- The second number \(B = 2^2 \times 3^2\)

### Step 2: Determine the highest power of each prime factor
- For the prime factor \(2\):
- In \(A\), the power of \(2\) is \(3\).
- In \(B\), the power of \(2\) is \(2\).
- The highest power of \(2\) is \(3\).

- For the prime factor \(3\):
- In both \(A\) and \(B\), the power of \(3\) is \(2\).
- The highest power of \(3\) is \(2\).

### Step 3: Write the LCM using the highest powers found
- The LCM is given by the product of the highest powers of all prime factors:
\[ \text{LCM} = 2^{\text{max}(3, 2)} \times 3^{\text{max}(2, 2)} = 2^3 \times 3^2 \]

### Step 4: Calculate the value of the LCM
- Now we can calculate the value:
\[ \text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72 \]

### Final Answer
The LCM of \(2^3 \times 3^2\) and \(2^2 \times 3^2\) is \(72\).

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