What is the geometric mean of the data 2, 4, 8, 16, 32 ?

Options

A. 2
B. 4
C. 8
D. 16

Explanation

To find the geometric mean of the data set \(2, 4, 8, 16, 32\), we can follow these steps:

### Step 1: Identify the numbers
The given data set is \(2, 4, 8, 16, 32\).

### Step 2: Multiply the numbers together
We need to calculate the product of all the numbers in the data set:
\[ 2 \times 4 \times 8 \times 16 \times 32 \]

### Step 3: Calculate the product
Let's calculate the product step by step:
- First, calculate \(2 \times 4 = 8\)
- Next, calculate \(8 \times 8 = 64\)
- Then, calculate \(64 \times 16 = 1024\)
- Finally, calculate \(1024 \times 32 = 32768\)

So, the product of the numbers is:
\[ 2 \times 4 \times 8 \times 16 \times 32 = 32768 \]

### Step 4: Take the nth root
Since there are 5 numbers in the data set, we will take the 5th root of the product:
\[ \text{Geometric Mean} = (32768)^{\frac{1}{5}} \]

### Step 5: Calculate the 5th root
To find \( (32768)^{\frac{1}{5}} \):
- We can express \(32768\) as \(2^{15}\) (since \(2^{15} = 32768\)).
- Therefore, we have:
\[ (2^{15})^{\frac{1}{5}} = 2^{15/5} = 2^3 = 8 \]

### Final Answer
Thus, the geometric mean of the data \(2, 4, 8, 16, 32\) is:
\[ \boxed{8} \]

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