Sure, let's solve the given problem step-by-step.
We are given the expression:
[tex]\[
M = \left[ \frac{(x + y)^2 - (x - y)^2}{x y} \right]^{\frac{1}{2}}
\][/tex]
### Step 1: Simplify the Numerator
First, we need to simplify the expression [tex]\((x + y)^2 - (x - y)^2\)[/tex]. Let's expand both squares:
[tex]\[
(x + y)^2 = x^2 + 2xy + y^2
\][/tex]
[tex]\[
(x - y)^2 = x^2 - 2xy + y^2
\][/tex]
Now, subtract the second expanded expression from the first:
[tex]\[
(x + y)^2 - (x - y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)
\][/tex]
Distribute the minus sign:
[tex]\[
(x^2 + 2xy + y^2) - x^2 + 2xy - y^2 = 2xy + 2xy = 4xy
\][/tex]
So, the numerator simplifies to:
[tex]\[
(x + y)^2 - (x - y)^2 = 4xy
\][/tex]
### Step 2: Simplify the Denominator
The denominator is given by [tex]\(xy\)[/tex].
### Step 3: Combine the Numerator and Denominator
Now, we substitute these simplifications back into our original expression:
[tex]\[
M = \left[ \frac{4xy}{xy} \right]^{\frac{1}{2}}
\][/tex]
The [tex]\(xy\)[/tex] in the numerator and the denominator cancel out:
[tex]\[
M = \left[ 4 \right]^{\frac{1}{2}}
\][/tex]
### Step 4: Final Calculation
The square root of 4 is:
[tex]\[
\sqrt{4} = 2
\][/tex]
Hence, the value of [tex]\(M\)[/tex] is:
[tex]\[
M = 2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
