The given mathematical expression in the image is:

$\frac{1}{\sqrt{x} - 2} - \frac{3}{x - 8}$

To simplify or solve this expression, the steps are as follows:

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Step 1: Find the common denominator

The denominators are $\sqrt{x} - 2$ and $x - 8$. Their least common denominator (LCD) is:

$(\sqrt{x} - 2)(x - 8)$

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Step 2: Rewrite each fraction with the common denominator

$\frac{1}{\sqrt{x} - 2} = \frac{x - 8}{(\sqrt{x} - 2)(x - 8)}$ $\frac{3}{x - 8} = \frac{3(\sqrt{x} - 2)}{(\sqrt{x} - 2)(x - 8)}$

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Step 3: Combine the fractions

Now that both fractions have the same denominator, combine the numerators:

$\frac{x - 8}{(\sqrt{x} - 2)(x - 8)} - \frac{3(\sqrt{x} - 2)}{(\sqrt{x} - 2)(x - 8)}$

$= \frac{(x - 8) - 3(\sqrt{x} - 2)}{(\sqrt{x} - 2)(x - 8)}$

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Step 4: Simplify the numerator

Expand and simplify the numerator:

$(x - 8) - 3(\sqrt{x} - 2) = x - 8 - 3\sqrt{x} + 6 = x - 2 - 3\sqrt{x}$

So the numerator becomes:

$x - 2 - 3\sqrt{x}$

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Final Expression

The simplified expression is:

$\frac{x - 2 - 3\sqrt{x}}{(\sqrt{x} - 2)(x - 8)}$

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If you would like me to explain any of the steps further or solve for specific values of $x$, feel free to ask!

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Related Questions:

1. How do we find the domain of this expression?
2. What happens when $\sqrt{x} = 2$ or $x = 8$?
3. How can this expression be simplified further for specific values of $x$?
4. What does this expression represent graphically?
5. Can this expression be rewritten without square roots in the numerator?

Tip:

Always check the domain restrictions for expressions involving square roots and denominators to avoid undefined values.