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Rewrite the following expression so that it is easier to read:
[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]

Sagot :

GinnyAnsweridn
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]
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