Alright, let's tackle the expression step-by-step.
We have the expression:
[tex]\[
\left[ \frac{x^m - y^m}{x^{m/2} - y^{m/2}} - \frac{x^m - y^m}{x^{m/2} + y^{m/2}} \right]^{-2}
\][/tex]
Step 1: Simplify each term
First, we'll look at the two fractions inside the brackets. We'll call the first term [tex]\(A\)[/tex] and the second term [tex]\(B\)[/tex]:
[tex]\[
A = \frac{x^m - y^m}{x^{m/2} - y^{m/2}}
\][/tex]
[tex]\[
B = \frac{x^m - y^m}{x^{m/2} + y^{m/2}}
\][/tex]
Step 2: Common factor for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]
Next, we recognize that both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] share the common numerator [tex]\(x^m - y^m\)[/tex].
Step 3: Simplify the expressions
For [tex]\(A\)[/tex]:
[tex]\[
A = \frac{x^m - y^m}{x^{m/2} - y^{m/2}}
\][/tex]
For [tex]\(B\)[/tex]:
[tex]\[
B = \frac{x^m - y^m}{x^{m/2} + y^{m/2}}
\][/tex]
To simplify the expression [tex]\( A - B \)[/tex], we need to find a common denominator:
[tex]\[
A - B = \frac{(x^m - y^m)}{x^{m/2} - y^{m/2}} - \frac{(x^m - y^m)}{x^{m/2} + y^{m/2}}
\][/tex]
The common denominator for these terms is [tex]\( (x^{m/2} - y^{m/2})(x^{m/2} + y^{m/2}) \)[/tex].
Step 4: Combine the fractions
[tex]\[
A - B = \frac{(x^m - y^m)(x^{m/2} + y^{m/2}) - (x^m - y^m)(x^{m/2} - y^{m/2})}{(x^{m/2} - y^{m/2})(x^{m/2} + y^{m/2})}
\][/tex]
[tex]\[
= \frac{(x^m - y^m)(x^{m/2} + y^{m/2}) - (x^m - y^m)(x^{m/2} - y^{m/2})}{x^m - y^m}
\][/tex]
Step 5: Simplify the numerator
[tex]\[
= \frac{(x^m - y^m)x^{m/2} + (x^m - y^m)y^{m/2} - (x^m - y^m)x^{m/2} + (x^m - y^m)y^{m/2}}{(x^{m/2} - y^{m/2})(x^{m/2} + y^{m/2})}
\][/tex]
Notice the terms [tex]\( (x^m - y^m)x^{m/2} \)[/tex] cancel out:
[tex]\[
= \frac{2(x^m - y^m)y^{m/2}}{(x^{m/2} - y^{m/2})(x^{m/2} + y^{m/2})}
\][/tex]
Since [tex]\( x^m - y^m = (x^{m/2} - y^{m/2})(x^{m/2} + y^{m/2}) \)[/tex]:
[tex]\[
= \frac{2y^{m/2}}{1}
\][/tex]
[tex]\[
= 2y^{m/2}
\][/tex]
Step 6: Raise the simplified expression to the power of [tex]\(-2\)[/tex]
Finally, we need to raise [tex]\(2y^{m/2}\)[/tex] to the power of [tex]\(-2\)[/tex]:
[tex]\[
\left(2y^{m/2}\right)^{-2} = \frac{1}{(2y^{m/2})^2} = \frac{1}{4y^m}
\][/tex]
Hence, the simplified expression is:
[tex]\[
\frac{1}{4y^m}
\][/tex]
So, the final answer is:
[tex]\[
\boxed{\frac{1}{4y^m}}
\][/tex]
