To simplify the expression \(\frac{1}{(1-\sqrt{x})}-\frac{1}{(1+\sqrt{x})}+\frac{\sqrt{x}}{(1-x)}\), we'll break it down into individual components and combine them step-by-step.
1. Consider the first term: \(\frac{1}{1 - \sqrt{x}}\)
2. Consider the second term: \(\frac{1}{1 + \sqrt{x}}\)
3. Combine the first two terms:
To combine these terms, let's find a common denominator.
[tex]\[
\frac{1}{1 - \sqrt{x}} - \frac{1}{1 + \sqrt{x}} = \frac{(1 + \sqrt{x}) - (1 - \sqrt{x})}{(1 - \sqrt{x})(1 + \sqrt{x})}
\][/tex]
Simplify the numerator:
[tex]\[
(1 + \sqrt{x}) - (1 - \sqrt{x}) = 1 + \sqrt{x} - 1 + \sqrt{x} = 2\sqrt{x}
\][/tex]
Simplify the denominator:
[tex]\[
(1 - \sqrt{x})(1 + \sqrt{x}) = 1 - x
\][/tex]
Thus, combining the first two terms gives:
[tex]\[
\frac{2\sqrt{x}}{1 - x}
\][/tex]
4. Consider the third term: \(\frac{\sqrt{x}}{1 - x}\)
5. Combine the result of the first two terms with the third term:
Since both terms have the same denominator, we can directly combine them:
[tex]\[
\frac{2\sqrt{x}}{1 - x} + \frac{\sqrt{x}}{1 - x} = \frac{2\sqrt{x} + \sqrt{x}}{1 - x}
\][/tex]
Combine the numerators:
[tex]\[
2\sqrt{x} + \sqrt{x} = 3\sqrt{x}
\][/tex]
6. Final result:
[tex]\[
\frac{3\sqrt{x}}{1 - x}
\][/tex]
Therefore, the simplified form of the expression \(\frac{1}{(1-\sqrt{x})}-\frac{1}{(1+\sqrt{x})}+\frac{\sqrt{x}}{(1-x)}\) is:
[tex]\[
\frac{3\sqrt{x}(1 - x)}{(1 - x)^2} = \frac{(3\sqrt{x})(1 - x)}{(1 - x)^2}
\][/tex]
Here, the expression \(\frac{3\sqrt{x}(1 - x)}{(1 - x)^2}\) simplifies to:
[tex]\[
\frac{3\sqrt{x}(1 - x)}{x^2 - 2x + 1}
\][/tex]
Thus, the complete and simplified expression is:
[tex]\[
\frac{3\sqrt{x}(1 - x)}{(1 - \sqrt{x})^2}
\][/tex]
The final simplified expression is:
[tex]\[
\frac{3\sqrt{x} (1 - x)}{(x^2 - 2x + 1)}
\][/tex]
