Find the product o three geometric means between 4 and 1/4.

Explanation

To find the product of three geometric means between 4 and \( \frac{1}{4} \), we can follow these steps:

### Step 1: Understand the problem
We need to find three geometric means (let's call them \( g_1, g_2, g_3 \)) that lie between the numbers 4 and \( \frac{1}{4} \). This means we have a sequence: \( 4, g_1, g_2, g_3, \frac{1}{4} \).

### Step 2: Use the property of geometric means
If \( a, g_1, g_2, g_3, b \) are in geometric progression (GP), then the product of the geometric means can be calculated using the formula:
\[ g_1 \cdot g_2 \cdot g_3 = \sqrt[3]{a \cdot b} \]
where \( a \) is the first term (4) and \( b \) is the last term (\( \frac{1}{4} \)).

### Step 3: Substitute the values
Substituting \( a = 4 \) and \( b = \frac{1}{4} \) into the formula gives:
\[ g_1 \cdot g_2 \cdot g_3 = \sqrt[3]{4 \cdot \frac{1}{4}} \]

### Step 4: Simplify the expression
Now, simplify the expression inside the cube root:
\[ 4 \cdot \frac{1}{4} = 1 \]
Thus, we have:
\[ g_1 \cdot g_2 \cdot g_3 = \sqrt[3]{1} \]

### Step 5: Calculate the cube root
The cube root of 1 is:
\[ \sqrt[3]{1} = 1 \]

### Conclusion
Therefore, the product of the three geometric means between 4 and \( \frac{1}{4} \) is:
\[ \boxed{1} \]