The mean proportional between 2 and 32 is _______.

Answer

To find the mean proportional between the numbers 2 and 32, we can follow these steps:

### Step 1: Understand the concept of mean proportional
The mean proportional \( x \) between two numbers \( a \) and \( b \) is defined such that the ratio \( \frac{a}{x} \) is equal to the ratio \( \frac{x}{b} \). This can be expressed mathematically as:
\[ \frac{a}{x} = \frac{x}{b} \]

### Step 2: Set up the equation
Given \( a = 2 \) and \( b = 32 \), we can set up the equation:
\[ \frac{2}{x} = \frac{x}{32} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 2 \cdot 32 = x \cdot x \]
This simplifies to:
\[ 64 = x^2 \]

### Step 4: Solve for \( x \)
To find \( x \), we take the square root of both sides:
\[ x = \sqrt{64} \]

### Step 5: Calculate the square root
Calculating the square root, we find:
\[ x = 8 \]

### Conclusion
Thus, the mean proportional between 2 and 32 is \( 8 \).

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