Cmmon factor of `27 a ^(3) b ^(2), 36 a ^(2) b^(3), 54 a ^(2) b ^(2)` is

Options

A. `9a ^(2) b ^(2)`
B. `18ab `
C. `108 a ^(2) b ^(2)`
D. `54 a ^(2) b ^(3)`

Answer

To find the common factor of the expressions \(27 a^{3} b^{2}\), \(36 a^{2} b^{3}\), and \(54 a^{2} b^{2}\), we will follow these steps:

### Step 1: Find the coefficients' GCF
First, we need to find the greatest common factor (GCF) of the numerical coefficients: 27, 36, and 54.

- **Factors of 27**: \(1, 3, 9, 27\)
- **Factors of 36**: \(1, 2, 3, 4, 6, 9, 12, 18, 36\)
- **Factors of 54**: \(1, 2, 3, 6, 9, 18, 27, 54\)

The common factors are \(1, 3, 9\). The greatest of these is **9**.

### Step 2: Find the common variable factors
Next, we will determine the common variable factors for \(a\) and \(b\).

- For \(a\):
- \(27 a^{3} b^{2}\) has \(a^{3}\)
- \(36 a^{2} b^{3}\) has \(a^{2}\)
- \(54 a^{2} b^{2}\) has \(a^{2}\)

The lowest power of \(a\) that is common in all three terms is \(a^{2}\).

- For \(b\):
- \(27 a^{3} b^{2}\) has \(b^{2}\)
- \(36 a^{2} b^{3}\) has \(b^{3}\)
- \(54 a^{2} b^{2}\) has \(b^{2}\)

The lowest power of \(b\) that is common in all three terms is \(b^{2}\).

### Step 3: Combine the GCFs
Now, we combine the GCF of the coefficients with the common variable factors:

\[ \text{Common Factor} = 9 a^{2} b^{2} \]

### Final Answer
Thus, the common factor of \(27 a^{3} b^{2}\), \(36 a^{2} b^{3}\), and \(54 a^{2} b^{2}\) is:

\[ \boxed{9 a^{2} b^{2}} \]