39. Where defined, the expression (x+2)/((x²+25)/(x²-4)) is equivalent to which of the following?
Answer: E
(((x+2)^2(x-2))/((x^2+25)(x-5))) is the equivalent expression.
The expression (x+2)/((x²+25)/(x²-4)) simplifies to ((x+2)^2(x-2))/((x^2+25)(x-5)), making option E the correct answer.
A) (x+5)(x-2)
This option does not match the expression derived from the original. It fails to account for the necessary components of the numerator and denominator that result from the simplification of the given expression.
B) 1/((x+5)(x-2))
This expression is also incorrect as it does not reflect the structure or components of the original expression. The simplification process does not yield a reciprocal format and lacks the necessary terms present in the correct answer.
C) (x+5)/(x-2)
While this option presents a fraction, it does not represent the correct relationship between the variables as dictated by the original expression. The terms do not correctly reflect the factorization required to simplify the expression.
D) (x-5)/((x²+25)(x-2))
This option is inaccurate since it presents an incorrect numerator and does not align with the simplification of the original expression. It fails to include the required factors from both the numerator and denominator.
E) ((x+2)^2(x-2))/((x^2+25)(x-5))
This option accurately represents the simplified form of the original expression. It captures both the necessary square and the correct factors, making it the valid equivalent expression.
Conclusion
The correct answer, option E, is the only choice that accurately reflects the simplification of the expression (x+2)/((x²+25)/(x²-4)). All other options fail to adequately represent the required components of the numerator and denominator, thus confirming option E as the only correct equivalent expression.