Find the geometric mean between :
`(4)/(9)` and `(9)/(4)`

Answer

To find the geometric mean between \( \frac{4}{9} \) and \( \frac{9}{4} \), we can follow these steps:

### Step 1: Understand the formula for geometric mean
The geometric mean (GM) of two numbers \( a \) and \( b \) is given by the formula:
\[ GM = \sqrt{a \cdot b} \]

### Step 2: Identify the values of \( a \) and \( b \)
In this case, we have:
- \( a = \frac{4}{9} \)
- \( b = \frac{9}{4} \)

### Step 3: Substitute the values into the formula
Now, we substitute \( a \) and \( b \) into the geometric mean formula:
\[ GM = \sqrt{\left(\frac{4}{9}\right) \cdot \left(\frac{9}{4}\right)} \]

### Step 4: Multiply the fractions
To multiply the fractions, we multiply the numerators and the denominators:
\[ \frac{4}{9} \cdot \frac{9}{4} = \frac{4 \cdot 9}{9 \cdot 4} = \frac{36}{36} = 1 \]

### Step 5: Take the square root
Now, we take the square root of the result:
\[ GM = \sqrt{1} = 1 \]

### Final Answer
Thus, the geometric mean between \( \frac{4}{9} \) and \( \frac{9}{4} \) is:
\[ \boxed{1} \]

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