The geometric mean of three numbers was conputed as 6. It was subsequently found that, in this computation, a number 8 was wrongly read as 12. What is the correct geometric mean ?

Options

A. 4
B. `root(3)(5)`
C. `2root(3)(18)`
D. None of these

Explanation

To solve the problem, we need to find the correct geometric mean after correcting the error in the computation. Let's break down the steps:

### Step 1: Understand the given information
We know that the geometric mean of three numbers was computed as 6. This means:
\[ \sqrt[3]{x_1 \cdot x_2 \cdot x_3} = 6 \]
Cubing both sides gives:
\[ x_1 \cdot x_2 \cdot x_3 = 6^3 = 216 \]

### Step 2: Identify the incorrect number
In the computation, one of the numbers (which we will denote as \(x_3\)) was incorrectly read as 12 instead of the correct value of 8. Therefore, we have:
\[ x_1 \cdot x_2 \cdot 12 = 216 \]

### Step 3: Solve for \(x_1 \cdot x_2\)
To find \(x_1 \cdot x_2\), we can rearrange the equation:
\[ x_1 \cdot x_2 = \frac{216}{12} = 18 \]

### Step 4: Calculate the correct geometric mean
Now, we need to find the correct geometric mean using the correct values. The correct geometric mean will be:
\[ \sqrt[3]{x_1 \cdot x_2 \cdot 8} \]
Substituting \(x_1 \cdot x_2 = 18\):
\[ \sqrt[3]{18 \cdot 8} = \sqrt[3]{144} \]

### Step 5: Simplify \(\sqrt[3]{144}\)
We can express 144 as:
\[ 144 = 2^4 \cdot 3^2 \]
Thus, we can rewrite the cube root:
\[ \sqrt[3]{144} = \sqrt[3]{2^4 \cdot 3^2} = \sqrt[3]{2^3 \cdot 2^1 \cdot 3^2} = 2 \cdot \sqrt[3]{2 \cdot 9} = 2 \cdot \sqrt[3]{18} \]

### Final Answer
The correct geometric mean is:
\[ 2 \cdot \sqrt[3]{18} \]

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