If x > 1, which of the following is equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)) ?

Rationale

sqrt(x - 1)/sqrt(x + 1) is equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)).

To simplify the expression sqrt(x - sqrt(x))/(x + sqrt(x)), we can factor both the numerator and denominator, leading to the equivalent expression of sqrt(x - 1)/sqrt(x + 1).

A (√(x - 1)/√(x - 1)) CORRECT

This expression is indeed equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)). By factoring, we find that the numerator simplifies to sqrt(x - 1) and the denominator to sqrt(x + 1), confirming their equivalence.

B (√(x - 1)/√(x + 1)) YOUR ANSWER

This choice contains the correct structure, but it is identical to the correct answer. Therefore, this option is misleading as it does not represent a different expression but rather the same one as option A.

C (√(x + 1)/√(x - 1)) YOUR ANSWER

This option is incorrect because it incorrectly reverses the terms in both the numerator and the denominator. The expressions are fundamentally different, leading to a different value than the original expression.

D (√(x + 1)/√(x + 1)) YOUR ANSWER

This choice simplifies to 1, which does not match the original expression. Therefore, it cannot be equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)) as that expression does not simplify to a constant.

E (√(x - 1)/√(x + 1)) YOUR ANSWER

While this option resembles the correct structure, it is simply a repetition of option A. As a result, it doesn't provide a different expression to evaluate.

Conclusion

The simplification of sqrt(x - sqrt(x))/(x + sqrt(x)) leads to the expression sqrt(x - 1)/sqrt(x + 1), confirming it as the only equivalent form among the choices. Options B and E are misleadingly identical to the correct answer, while C and D represent fundamentally different values that do not satisfy the original expression.

If x > 1, which of the following is equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)) ?

sqrt(x - 1)/sqrt(x + 1) is equivalent to sqrt(x - sqrt(x))/(x + sqrt(x)).

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