To solve the given problem, we need to rationalize the denominator of the fraction:
[tex]\[
\frac{1}{\sqrt{x}-\sqrt{x-1}}
\][/tex]
Rationalizing the denominator involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\( \sqrt{x} - \sqrt{x-1} \)[/tex] is [tex]\( \sqrt{x} + \sqrt{x-1} \)[/tex]. Let's proceed with this process:
1. Multiply by the Conjugate:
[tex]\[
\frac{1}{\sqrt{x}-\sqrt{x-1}} \times \frac{\sqrt{x} + \sqrt{x-1}}{\sqrt{x} + \sqrt{x-1}}
\][/tex]
2. Simplify the Numerator:
The numerator now becomes:
[tex]\[
1 \times (\sqrt{x} + \sqrt{x-1}) = \sqrt{x} + \sqrt{x-1}
\][/tex]
3. Simplify the Denominator:
The denominator now becomes:
[tex]\[
(\sqrt{x} - \sqrt{x-1}) \times (\sqrt{x} + \sqrt{x-1})
\][/tex]
Using the difference of squares formula, [tex]\(a^2 - b^2 = (a-b)(a+b)\)[/tex], we have:
[tex]\[
(\sqrt{x})^2 - (\sqrt{x-1})^2
\][/tex]
Simplifying further:
[tex]\[
x - (x-1) = x - x + 1 = 1
\][/tex]
4. Combine the Results:
After simplifying both the numerator and the denominator, the fraction becomes:
[tex]\[
\frac{\sqrt{x} + \sqrt{x-1}}{1}
\][/tex]
So the simplified form of the given fraction is:
[tex]\[
\sqrt{x} + \sqrt{x-1}
\][/tex]
Thus, the equivalent choice is:
D. [tex]\( \sqrt{x} + \sqrt{x-1} \)[/tex]
