Question 15

√(x - 1)/√(x - 1)

The expression √(x - sqrt(x))/(x + sqrt(x)) simplifies to √(x - 1)/√(x - 1) when rationalizing the numerator and denominator. This demonstrates that the two sides are equivalent.

A) √(x - 1)/√(x - 1)

This option is correct because it simplifies directly from the given expression. By factoring out common terms and using the properties of square roots, the original expression can be rewritten as this option, thereby confirming its validity.

B) √(x - 1)/√(x + 1)

This option is incorrect because it does not represent the simplification of the original expression. The denominator in this case introduces an additional term that is not present in the original, leading to an entirely different expression.

C) √(x + 1)/√(x - 1)

This option is also incorrect. It swaps the components of the numerator and denominator in a way that does not correspond to the simplification process of the original expression, leading to a fundamentally different result.

D) √(x + 1)/√(x + 1)

This option is incorrect as well. Similar to option C, it introduces terms that are not present in the original expression, resulting in an entirely different value when evaluated.

E) √(x - 1)/√(x + 1)

This option is incorrect because it retains the numerator correctly but alters the denominator. This modification changes the equivalence of the expression, making it not match the original form.

Conclusion

The correct option, √(x - 1)/√(x - 1), is the only choice that accurately reflects the simplification of the original expression. All other options either introduce new terms or alter the existing structure, resulting in expressions that do not equate to the original expression when evaluated. Thus, A stands out as the definitive correct answer.