Find the geometric mean between :
14 and `(7)/(32)`

Answer

To find the geometric mean between the numbers 14 and \( \frac{7}{32} \), we can follow these steps:

### Step 1: Identify the two numbers
Let \( A = 14 \) and \( B = \frac{7}{32} \).

### Step 2: Use the formula for the geometric mean
The formula for the geometric mean (GM) of two numbers \( A \) and \( B \) is given by:
\[ GM = \sqrt{A \times B} \]

### Step 3: Calculate \( A \times B \)
Now, let's calculate \( A \times B \):
\[ A \times B = 14 \times \frac{7}{32} \]
To perform this multiplication:
\[ A \times B = \frac{14 \times 7}{32} = \frac{98}{32} \]

### Step 4: Simplify \( \frac{98}{32} \)
Next, we simplify \( \frac{98}{32} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
\[ \frac{98 \div 2}{32 \div 2} = \frac{49}{16} \]

### Step 5: Find the square root
Now, we need to find the square root of \( \frac{49}{16} \):
\[ GM = \sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4} \]

### Final Answer
Thus, the geometric mean between 14 and \( \frac{7}{32} \) is:
\[ \frac{7}{4} \]

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