Let's determine the product of the given rational expressions:
[tex]\[
\frac{x-8}{x+11} \cdot \frac{x+8}{x-11}
\][/tex]
Step-by-Step Solution:
1. Multiply the Numerators:
We start by multiplying the numerators of the two fractions:
[tex]\[
(x - 8) \cdot (x + 8)
\][/tex]
2. Multiply the Denominators:
Next, we multiply the denominators of the two fractions:
[tex]\[
(x + 11) \cdot (x - 11)
\][/tex]
3. Form the Product of the Rational Expressions:
Combining the results of steps 1 and 2, the product of the two rational expressions is:
[tex]\[
\frac{(x - 8)(x + 8)}{(x + 11)(x - 11)}
\][/tex]
4. Simplify the Expression:
To simplify, we recognize that the numerators and denominators are products of binomials that can be expressed as the difference of squares:
[tex]\[
(x - 8)(x + 8) = x^2 - 64
\][/tex]
[tex]\[
(x + 11)(x - 11) = x^2 - 121
\][/tex]
Therefore, the simplified form of the product is:
[tex]\[
\frac{x^2 - 64}{x^2 - 121}
\][/tex]
5. Conclusion:
From the options given:
- A. [tex]\(\frac{x^2-121}{x^2-64}\)[/tex]
- B. [tex]\(\frac{x^2-64}{x^2}\)[/tex]
- C. [tex]\(\frac{64}{121}\)[/tex]
- D. [tex]\(\frac{x^2-64}{x^2-121}\)[/tex]
The correct answer is:
[tex]\[
\boxed{\frac{x^2 - 64}{x^2 - 121}}
\][/tex] Which corresponds to option D.
