To rationalize the denominator of the expression [tex]\(\frac{10}{\sqrt{x} - \sqrt{x - 5}}\)[/tex] when [tex]\(x \geq 5\)[/tex], follow these steps:
1. Identify the denominator:
[tex]\(\sqrt{x} - \sqrt{x - 5}\)[/tex]
2. Multiply the numerator and denominator by the conjugate of the denominator:
The conjugate of [tex]\(\sqrt{x} - \sqrt{x - 5}\)[/tex] is [tex]\(\sqrt{x} + \sqrt{x - 5}\)[/tex].
Hence, we multiply the numerator and denominator by [tex]\(\sqrt{x} + \sqrt{x - 5}\)[/tex]:
[tex]\[
\frac{10}{\sqrt{x} - \sqrt{x - 5}} \cdot \frac{\sqrt{x} + \sqrt{x - 5}}{\sqrt{x} + \sqrt{x - 5}}
\][/tex]
3. Simplify the expression:
The numerator becomes:
[tex]\[
10 \cdot (\sqrt{x} + \sqrt{x - 5}) = 10 (\sqrt{x} + \sqrt{x - 5})
\][/tex]
The denominator becomes:
[tex]\[
(\sqrt{x} - \sqrt{x - 5})(\sqrt{x} + \sqrt{x - 5})
\][/tex]
Using the difference of squares formula, [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], where [tex]\(a = \sqrt{x}\)[/tex] and [tex]\(b = \sqrt{x - 5}\)[/tex], we get:
[tex]\[
(\sqrt{x})^2 - (\sqrt{x - 5})^2 = x - (x - 5) = x - x + 5 = 5
\][/tex]
Thus, the denominator simplifies to 5.
4. Final simplified form:
Now, the entire expression simplifies to:
[tex]\[
\frac{10 (\sqrt{x} + \sqrt{x - 5})}{5} = 2 (\sqrt{x} + \sqrt{x - 5})
\][/tex]
Therefore, the equivalent expression with a rationalized denominator is:
[tex]\[
\boxed{2(\sqrt{x} + \sqrt{x - 5})}
\][/tex]
So the correct answer is:
A. [tex]\(2(\sqrt{x} + \sqrt{x - 5})\)[/tex]
