Geometric mean of 2, 6, 18, 54is

Options

A. 12
B. `13sqrt(3)`
C. `sqrt(8xx24)`
D. none of these

Answer

To find the geometric mean of the numbers 2, 6, 18, and 54, we can follow these steps:

### Step 1: Understand the formula for geometric mean
The geometric mean (GM) of n numbers is given by the formula:
\[ GM = \sqrt[n]{x_1 \times x_2 \times x_3 \times ... \times x_n} \]
where \(x_1, x_2, x_3, ..., x_n\) are the numbers.

### Step 2: Identify the numbers
In this case, the numbers are:
- \(x_1 = 2\)
- \(x_2 = 6\)
- \(x_3 = 18\)
- \(x_4 = 54\)

### Step 3: Multiply the numbers
Now, we will multiply these numbers together:
\[ 2 \times 6 \times 18 \times 54 \]

### Step 4: Calculate the product
Calculating step-by-step:
1. \(2 \times 6 = 12\)
2. \(12 \times 18 = 216\)
3. \(216 \times 54 = 11664\)

So, the product is \(11664\).

### Step 5: Find the fourth root
Now, we need to find the fourth root of \(11664\):
\[ GM = \sqrt[4]{11664} \]

### Step 6: Simplify the fourth root
To simplify \(11664\), we can factor it:
\[ 11664 = 2^2 \times 3^6 \]
Now, we can take the fourth root:
\[ GM = \sqrt[4]{2^2 \times 3^6} = \sqrt[4]{2^2} \times \sqrt[4]{3^6} \]
\[ = 2^{2/4} \times 3^{6/4} = 2^{1/2} \times 3^{3/2} = \sqrt{2} \times 3\sqrt{3} = 3\sqrt{6} \]

### Final Answer
Thus, the geometric mean of 2, 6, 18, and 54 is:
\[ GM = 6\sqrt{3} \]