Geometric Mean Calculation for X

The geometric mean (GM) of a set of $n$ numbers is the $n$-th root of the product of the numbers. The formula is:

$ GM = \sqrt[n]{a_1 \times a_2 \times ... \times a_n} $

In this problem, we have four numbers: 2, X, 16, and 32. The geometric mean is given as 8.

Solving for X

1. Set up the geometric mean equation using the given values:

   $ 8 = \sqrt[4]{2 \times X \times 16 \times 32} $

2. To eliminate the fourth root, raise both sides of the equation to the power of 4:

   $ 8^4 = 2 \times X \times 16 \times 32 $

3. Calculate the values:
   • $8^4 = 4096$
   • $2 \times 16 \times 32 = 32 \times 32 = 1024$
   The equation becomes:

   $ 4096 = X \times 1024 $

4. Solve for X by dividing both sides by 1024:

   $ X = \frac{4096}{1024} $

   $ X = 4 $

Therefore, the value of X is 4.