Mean Proportional Calculation for 324 and 16

The question asks us to find the mean proportional of the numbers 324 and 16. The mean proportional, also known as the geometric mean, between two positive numbers $a$ and $b$ is given by the formula:

$ \text{Mean Proportional} = \sqrt{a \times b} $

Understanding Mean Proportional

The mean proportional helps us find a number that maintains the same ratio between the first number and the mean proportional, as the ratio between the mean proportional and the second number. Essentially, it's the square root of the product of the two numbers.

Step-by-Step Solution

1. Identify the numbers: The two given numbers are $a = 324$ and $b = 16$.
2. Apply the formula: Substitute these values into the mean proportional formula: $ \text{Mean Proportional} = \sqrt{324 \times 16} $
3. Calculate the product: First, find the product of 324 and 16: $ 324 \times 16 = 5184 $
4. Find the square root: Now, calculate the square root of the product: $ \text{Mean Proportional} = \sqrt{5184} $
5. Simplify the square root: We can simplify this by finding the square root of each number individually before multiplying, or by finding the square root of the combined product. Alternatively, we can break it down: $ \sqrt{324 \times 16} = \sqrt{324} \times \sqrt{16} $ We know that $\sqrt{324} = 18$ (since $18 \times 18 = 324$) and $\sqrt{16} = 4$ (since $4 \times 4 = 16$). So, the calculation becomes: $ \text{Mean Proportional} = 18 \times 4 $
6. Final Calculation: $ 18 \times 4 = 72 $

Therefore, the mean proportional of 324 and 16 is 72.