Geometric mean of 3, 9 and 27, is

Options

A. 18
B. 6
C. 9
D. none of these

Answer

To find the geometric mean of the numbers 3, 9, and 27, we can follow these steps:

### Step 1: Understand the formula for geometric mean
The geometric mean (GM) of \( n \) numbers \( x_1, x_2, x_3, \ldots, x_n \) is given by the formula:
\[ GM = \sqrt[n]{x_1 \times x_2 \times x_3 \times \ldots \times x_n} \]
In this case, since we have 3 numbers, the formula simplifies to:
\[ GM = \sqrt[3]{x_1 \times x_2 \times x_3} \]

### Step 2: Identify the numbers
The numbers given are:
- \( x_1 = 3 \)
- \( x_2 = 9 \)
- \( x_3 = 27 \)

### Step 3: Multiply the numbers
Now, we need to calculate the product of these numbers:
\[ 3 \times 9 \times 27 \]

### Step 4: Simplify the multiplication
We can simplify the multiplication step-by-step:
1. First, multiply \( 3 \) and \( 9 \):
\[ 3 \times 9 = 27 \]
2. Now, multiply the result by \( 27 \):
\[ 27 \times 27 = 729 \]

### Step 5: Calculate the cube root
Now that we have the product \( 729 \), we need to find the cube root:
\[ GM = \sqrt[3]{729} \]

### Step 6: Simplify the cube root
We know that:
\[ 729 = 9^3 \]
Thus, the cube root of \( 729 \) is:
\[ \sqrt[3]{729} = 9 \]

### Final Answer
Therefore, the geometric mean of the numbers \( 3, 9, \) and \( 27 \) is:
\[ \boxed{9} \]