If `M_(g,x)` is the geometric mean of N x's and `M_(g,y)` is the geometric mean of N y's, then the grometric mean `M_(g)` of the 2N values is

Options

A. `Nsqrt(M_(g,x)M_(g,y))`
B. `sqrt(M_(g,x)M_(g,y))`
C. `(M_(g,x)M_(g,y))`
D. `(M_(g,x)M_(g,y))^(2)`

Answer

To solve the problem, we need to find the geometric mean \( M_g \) of \( 2N \) values, given that \( M_{g,x} \) is the geometric mean of \( N \) values \( x_1, x_2, \ldots, x_N \) and \( M_{g,y} \) is the geometric mean of \( N \) values \( y_1, y_2, \ldots, y_N \).

### Step-by-step Solution:

1. **Define the Geometric Means**:
The geometric mean of \( N \) values \( x_1, x_2, \ldots, x_N \) is given by:
\[ M_{g,x} = (x_1 \cdot x_2 \cdot \ldots \cdot x_N)^{\frac{1}{N}} \]
Similarly, for the \( y \) values:
\[ M_{g,y} = (y_1 \cdot y_2 \cdot \ldots \cdot y_N)^{\frac{1}{N}} \]

2. **Express the Products in Terms of Geometric Means**:
From the definitions above, we can express the products of the \( x \) and \( y \) values as:
\[ x_1 \cdot x_2 \cdot \ldots \cdot x_N = M_{g,x}^N \]
\[ y_1 \cdot y_2 \cdot \ldots \cdot y_N = M_{g,y}^N \]

3. **Combine the Values**:
The combined geometric mean \( M_g \) of the \( 2N \) values \( x_1, x_2, \ldots, x_N, y_1, y_2, \ldots, y_N \) is given by:
\[ M_g = (x_1 \cdot x_2 \cdot \ldots \cdot x_N \cdot y_1 \cdot y_2 \cdot \ldots \cdot y_N)^{\frac{1}{2N}} \]

4. **Substituting the Products**:
Substitute the expressions for the products from step 2 into the equation for \( M_g \):
\[ M_g = (M_{g,x}^N \cdot M_{g,y}^N)^{\frac{1}{2N}} \]

5. **Simplifying the Expression**:
Using the property of exponents, we can simplify:
\[ M_g = (M_{g,x} \cdot M_{g,y})^{\frac{N}{2N}} = (M_{g,x} \cdot M_{g,y})^{\frac{1}{2}} \]

6. **Final Result**:
Thus, the geometric mean \( M_g \) of the \( 2N \) values is:
\[ M_g = \sqrt{M_{g,x} \cdot M_{g,y}} \]